DYNAMIC  SYMMETRY 


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PUBLISHED  UNDER  THE  AUSPICES  OF 
THE  SCHOOL  OF  THE  FINE  ARTS,  YALE  UNIVERSITY, 
ON  THE  FOUNDATION  ESTABLISHED 
IN  MEMORY  OF 
RUTHERFORD  TROWBRIDGE 


AN  UNUSUALLY  HANDSOME  NOLAN  AMPHORA, 
FOGG  MUSEUM  AT  HARVARD 

A  theme  in  root-two 


DYNAMIC  SYMMETRY 

THE  GREEK  VASE 


BY  JAY  HAMBIDGE 


MCMXX  •  YALE  UNIVERSITY  PRESS 

NEW  HAVEN  CONNECTICUT  AND  NEW  YORK  CITY 

LONDON  •  HUMPHREY  MILFORD  •  OXFORD  UNIVERSITY  PRESS 


COPYRIGHT,  1920,  BY 
YALE  UNIVERSITY  PRESS 


THE  RUTHERFORD  TROWBRIDGE 
MEMORIAL  PUBLICATION 
FUND 


~^HE  present  volume  is  the  first  work  published  on  the  Ruther- 
ford Trowbridge  Memorial  Publication  Fund.  This  Founda- 
tion was  established  in  May,  1920,  through  a  gift  to  Yale 
University  by  his  widow  in  memory  of  Rutherford  Trowbridge,  Esq., 
of  New  Haven,  who  died  December  18,  191 8,  and  who  in  1899  had 
established  the  Thomas  Rutherford  Trowbridge  Memorial  Lectureship 
Fund  in  the  School  of  the  Fine  Arts  at  Yale.  It  was  in  a  series  of 
lectures  delivered  on  this  Foundation  that  the  material  comprised  in 
this  volume  was  first  given  to  the  public.  By  the  establishment  of 
the  Rutherford  Trowbridge  Memorial  Publication  Fund,  the  Univer- 
sity has  been  enabled  to  make  available  for  a  much  wider  audience 
the  work  growing  out  of  lectures  given  at  Yale  through  the  generosity 
of  one  who  sought  always  to  render  service  to  the  community  in  which 
he  lived;  and,  through  its  university,  to  the  world. 


ACKNOWLEDGMENT 


r 


""^0  the  School  of  the  Fine  Arts  of  Yale  University  credit  is 
due  for  this  book  on  the  shape  of  the  Greek  vase.  When  the 
discovery  was  made  that  the  design  forms  of  this  pottery  were 
strictly  dynamic  and  it  became  apparent  that  an  analysis  of 
a  sufficient  number  of  vase  examples  would  be  equivalent  to 
the  recovery  of  the  technical  methods  of  Greek  designers  of  the  classic  age, 
William  Sergeant  Kendall,  Dean  of  the  Yale  School  of  the  Fine  Arts,  imme- 
diately recognized  its  importance  and  offered  his  personal  service  and  that  of 
the  University  to  help  in  the  arduous  task  of  gathering  reliable  material  for  a 
volume. 

After  the  investigation  of  the  shape  of  the  Greek  vase  was  begun  the  two  great 
American  Museums,  the  Museum  of  Fine  Arts  of  Boston  and  the  Metropolitan 
Museum  of  New  York  City,  through  the  curators  of  their  departments  of 
Greek  art,  Dr.  L.  D.  Caskey  and  Miss  G.  M.  A.  Richter,  volunteered  their 
services  in  furthering  the  work.  Most  of  the  vase  examples  in  the  book  were 
measured  and  drawn  by  the  staffs  of  these  Museums,  which  readily  gave  per- 
mission for  their  publication.  Dr.  Caskey,  during  the  past  year,  has  devoted 
almost  all  his  time  to  a  critical  examination  of  the  entire  collection  of  the 
Greek  vases  in  the  Museum  of  Fine  Arts  with  the  result  that  he  now  has  a  book 
nearly  ready  for  publication.  He  is  especially  equipped  for  a  research  of  this 
character,  because  of  the  fact  that  in  addition  to  his  attainments  as  a  Greek 
scholar,  he  has  had  much  archaeological  experience  in  the  field  at  Athens  and 
elsewhere. 

JAY  HAMBIDGE 


TO  M.  L.  C. 


CONTENTS 

VITRUVIUS  ON  GREEK  SYMMETRY   9 

PREDICTION  BY  EDMOND  POTTIER  IN  1906  RELATIVE  TO  GREEK 
SYMMETRY..   10 

CHAPTER  ONE:  THE  BASIS  OF  DESIGN  IN  NATURE   11 

CHAPTER  TWO:  THE  ROOT  RECTANGLES   19 

CHAPTER  THREE :  THE  LEAF   30 

CHAPTER  FOUR:  ROOT  RECTANGLES  AND  SOME  VASE  FORMS   44 

CHAPTER  FIVE:  PLATO'S  MOST  BEAUTIFUL  SHAPE   59 

CHAPTER  SIX:  A  BRYGOS  KANTHAROS  AND  OTHER  POTTERY  EX- 
AMPLES OF  SIMILAR  RECTANGLE  SHAPES   65 

CHAPTER  SEVEN:  A  HYDRIA,  A  STAMNOS,  A  PYXIS  AND  OTHER 
VASE  FORMS     75 

CHAPTER  EIGHT:  FURTHER  ANALYSES  OF  VASE  FORMS   91 

CHAPTER  NINE:  SKYPHOI   105 

CHAPTER  TEN:  KYLIKES   114 

CHAPTER  ELEVEN:  VASE  ANALYSES,  CONTINUED   123 

CHAPTER  TWELVE:  STATIC  SYMMETRY   138 


LIST  OF  PLATES 

AN  UNUSUALLY  HANDSOME  NOLAN  AMPHORA,  FOGG  MUSEUM  AT 
HARVARD 

A  WHITE-GROUND  PYXIS,  MUSEUM  OF  FINE  ARTS,  BOSTON 

A  WHITE-GROUND  PYXIS,  METROPOLITAN  MUSEUM,  NEW  YORK 

AN  EARLY  BLACK-FIGURED  KYLIX  OF  UNUSUAL  DISTINCTION,  BOS- 
TON MUSEUM  OF  FINE  ARTS 

A  BLACK-FIGURED  HYDRIA,  MUSEUM  OF  FINE  ARTS,  BOSTON 

KANTHAROS,  CONSIDERED  BY  THE  WRITER  AS  ONE  OF  THE  FINEST 
OF  GREEK  CUPS 

LARGE  BRONZE  HYDRIA,  METROPOLITAN  MUSEUM,  NEW  YORK 

A  LARGE  STAMNOS,  METROPOLITAN  MUSEUM,  NEW  YORK 

A  DINOS  AND  STAND,  MUSEUM  OF  FINE  ARTS,  BOSTON 

A  BLACK-FIGURED  AMPHORA  FROM  THE  BOSTON  MUSEUM 

A  LARGE  BELL  KRATER  WITH  LUG  HANDLES,  MUSEUM  OF  FINE  ARTS, 
BOSTON 

A  RED-FIGURED  KALPIS  IN  THE  METROPOLITAN  MUSEUM,  NEW 
YORK 

A  BLACK-FIGURED  SKYPHOS,  METROPOLITAN  MUSEUM,  NEW  YORK 

A  BLACK-FIGURED  EYE  KYLIX,  MUSEUM  OF  FINE  ARTS,  BOSTON 

AN  EARLY  BLACK-FIGURED  LEKYTHOS,  STODDARD  COLLECTION  AT 
YALE 

A  BLACK  GLAZE  OINOCHOE  FROM  THE  STODDARD  COLLECTION  AT 
YALE 


FOREWORD 

SOME  twenty  years  ago,  the  writer,  being  impressed  by  the  inco- 
herence of  modern  design  and  convinced  that  there  must  exist  in 
nature  some  correlating  principle  which  could  give  artists  a  con- 
|  trol  of  areas,  undertook  a  comparative  study  of  the  bases  of  all 
design,  both  in  nature  and  in  art.  This  labor  resulted  in  the  de- 
termination of  two  types  of  symmetry  or  proportion,  one  of  which  possessed 
qualities  of  activity,  the  other  of  passivity.  For  convenience,  the  active  type 
was  termed  dynamic  symmetry,  the  other,  static  symmetry.  It  was  found  that 
the  passive  was  the  type  which  was  employed  most  naturally  by  artists,  either 
consciously  or  unconsciously;  in  fact,  no  design  which  would  be  recognized  as 
such — unless,  indeed,  it  were  dynamic — would  be  possible  without  the  use,  in 
some  degree,  of  this  passive  or  static  type.  It  is  apparent  in  nature  in  certain 
crystal  forms,  radiolaria,  diatoms,  flowers  and  seed  pods,  and  has  been  used 
consciously  in  art  at  several  periods. 

The  principle  of  dynamic  symmetry  is  manifest  in  shell  growth  and  in  leaf 
distribution  in  plants.  A  study  of  the  basis  of  design  in  art  shows  that  this  active 
symmetry  was  known  to  but  two  peoples,  the  Egyptians  and  the  Greeks;  the 
latter  only  having  developed  its  full  possibilities  for  purposes  of  art.  The  writer 
believes  that  he  has  now  recovered,  through  study  of  natural  form  and  shapes 
in  Greek  and  Egyptian  art,  this  principle  for  the  proportioning  of  areas. 

As  static  symmetry  is  more  or  less  known  and  its  principles  easily  under- 
stood, its  explanation  will  be  reserved  for  a  chapter  at  the  end  of  this  book.  Dy- 
namic symmetry,  on  the  contrary,  is  entirely  unrecognized  in  modern  times.  It  is 
more  subtle  and  more  vital  than  static  symmetry  and  is  pre-eminently  the  form 
to  be  employed  by  the  artist,  architect  and  craftsman.  After  an  explanation  of 
the  fundamental  principles  of  this  method  of  proportioning  spaces,  the  writer 
will  attempt  a  complete  exposition  of  its  application  in  art  through  analyses  of 
specific  examples  of  Greek  design.  He  believes  that  nothing  better  can  be  found 
for  this  purpose  than  Greek  pottery,  inasmuch  as  it  is  the  only  pottery  which 
is  absolutely  architectural  in  all  its  elements.  There  is  no  essential  difference 
between  the  plan  of  a  Greek  vase  and  the  plan  of  a  Greek  temple  or  theater, 
either  in  general  aspect,  or  in  detail.  The  curves  found  in  Greek  pottery  are 
identical  with  the  curves  of  mouldings  found  in  Greek  temples.  There  are  com- 
paratively few  temples  and  theaters,  while  there  are  many  thousands  of  vases, 
many  of  these  being  perfectly  preserved.  Other  reliable  material  for  study  is 
furnished  by  the  bas-reliefs  of  Egypt,  many  of  which,  like  the  vases  of  Greece, 
are  still  intact. 

The  history  of  dynamic  symmetry  may  be  given  in  a  few  words:  at  a  very 


s 


DYNAMIC  SYMMETRY 


early  date,  possibly  three  or  four  thousand  years  B.  C,  the  Egyptians  devel- 
oped an  empirical  scheme  for  surveying  land.  This  primitive  scheme  was  born 
of  necessity,  because  the  annual  overflow  of  the  Nile  destroyed  property  bound- 
aries. To  avoid  disputes  and  to  insure  an  equitable  taxation,  these  had  to  be  re- 
established; and  of  necessity,  also,  the  method  of  surveying  had  to  be  practica- 
ble and  simple.  It  required  but  two  men  and  a  knotted  rope. 

When  temple  and  tomb  building  began,  it  became  necessary  to  establish  a 
right  angle  and  lay  out  a  full  sized  plan  on  the  ground.  The  right  angle  was 
determined  by  marking  off"  twelve  units  on  the  rope,  four  of  these  units  forming 
one  side,  three  the  other,  and  five  the  hypotenuse  of  the  triangle,  a  method 
which  has  persisted  to  our  day.  This  was  the  origin  of  the  historic  "cording  of 
the  temple."2  From  this  the  step  to  the  formation  of  rectangular  plans  was 
simple.  From  the  larger  operation  of  surveying,  and  fixing  the  ground  plans  of 
buildings  by  the  power  which  the  right  angle  gave  toward  the  defining  of  ratio- 
relationship,  it  was  a  simple  matter  to  extend  and  adapt  this  method  to  the 
elevation  plan  and  the  detail  of  ornament,  in  short,  to  design  in  general,  to  the 
end  that  the  architect,  the  artist  or  the  craftsman  might  be  able  to  control  the 
proportioning  and  the  spacing  problems  involved  in  the  construction  of  build- 
ings as  well  as  those  of  pictorial  composition,  hieroglyphic  writing  and  decora- 
tion. At  some  time  during  the  Sixth  or  Seventh  Century  B.  C.  the  Greeks  ob- 
tained from  Egypt  knowledge  of  this  manner  of  correlating  elements  of  design. 
In  their  hands  it  was  highly  perfected  as  a  practical  geometry,  and  for  about 
three  hundred  years  it  provided  the  basic  principle  of  design  for  what  the 
writer  considers  the  finest  art  of  the  Classic  period.  Euclidean  geometry  gives 
us  the  Greek  development  of  the  idea  in  pure  mathematics;  but  the  secret  of 
its  artistic  application  completely  disappeared.  Its  recovery  has  given  us  dy- 
namic symmetry — a  method  of  establishing  the  relationship  of  areas  in  design- 
composition. 


VITRUVIUS  ON  GREEK  SYMMETRY3 


'  "^^HE  several  parts  which  constitute  a  temple  ought  to  be  sub- 
ject to  the  laws  of  symmetry;  the  principles  of  which  should 
be  familiar  to  all  who  profess  the  science  of  architecture. 
Symmetry  results  from  proportion,  which,  in  the  Greek  lan- 
guage, is  termed  analogy.  Proportion  is  the  commensuration 
of  the  various  constituent  parts  with  the  whole,  in  the  existence  of  which 
symmetry  is  found  to  consist.  For  no  building  can  possess  the  attributes  of 
composition  in  which  symmetry  and  proportion  are  disregarded;  nor  unless 
there  exists  that  perfect  conformation  of  parts  which  may  be  observed  in  a 
well-formed  human  being.  ...  Since,  therefore,  the  human  frame 
appears  to  have  been  formed  with  such  propriety  that  the  several  members 
are  commensurate  with  the  whole,  the  artists  of  antiquity  must  be  allowed 
to  have  followed  the  dictates  of  a  judgment  the  most  rational,  when,  trans- 
ferring to  the  works  of  art,  principles  derived  from  nature,  every  part  was  so 
regulated  as  to  bear  a  just  proportion  to  the  whole.  Now,  although  these 
principles  were  universally  acted  upon,  yet  they  were  more  particularly  at- 
tended to  in  the  construction  of  temples  and  sacred  edifices — the  beauties  or 
defects  of  which  were  destined  to  remain  as  a  perpetual  testimony  of  their 
skill  or  of  their  inability." 


PREDICTION  BY  EDMOND  POTTIER 
IN  1906  RELATIVE  TO  GREEK 
SYMMETRY 


WILL  add  that  the  proportions  of  the  vases,  the  relations  of  dimen- 
sions between  the  different  parts  of  the  vessel,  seem  among  the  Greeks 
to  have  been  the  object  of  minute  and  delicate  researches.  We  know  of 
cups  from  the  same  factory,  which,  while  similar  in  appearance,  are 
none  the  less  different  in  slight,  but  appreciable,  variations  of  structure 
{cf.,  for  example,  Furtwangler  and  Reichhold,  "Griechische  J^asenmalerei" 
p.  250).  One  might  perhaps  find  in  them,  if  one  made  a  profound  study  of  the 
subject,  a  system  of  measurement  analogous  to  that  of  statuary.  We  have,  in 
fact,  seen  that  at  its  origin  the  vase  is  not  to  be  separated  from  the  figurine 
(p.  78);  down  to  the  classical  period  it  retains  points  of  similarity  with  the 
structure  of  the  human  body  (Salle  H).  As  M.  Froehner  has  well  shown  in 
an  ingenious  article  {Revue  des  Deux  Mondes  1873,  c-  CIV,  p.  223),  we  our- 
selves speak  of  the  foot,  the  neck,  the  body,  the  lip  of  a  vase,  assimilating  the 
pottery  to  the  human  figure.  What,  then,  would  be  more  natural  than  to  sub- 
mit it  to  a  sort  of  plastic  canon,  which,  while  modified  in  the  course  of  time, 
would  be  based  on  simple  and  logical  rules?  I  have  remarked  ^Monuments 
Piot,  IX,"  p.  138)  that  the  maker  of  the  vase  of  Cleomenes  observed  a  rule 
illustrated  by  many  pieces  of  pottery  of  this  class,  when  he  made  the  height 
of  the  object  exactly  equal  to  its  width.  M.  Reichhold  (1.  c.  p.  181)  also  notes 
that  in  an  amphora  attributed  to  Euthymides  the  circumference  of  the  body 
is  exactly  double  the  height  of  the  vase.  I  believe  that  a  careful  examination  of 
the  subject  would  lead  to  interesting  observations  on  what  might  be  called 
the  "geometry  of  Greek  ceramics."  E.  Pottier,  Musee  National  du  Louvre, 
"Vases  antiques  III,"  p.  659. 


CHAPTER  ONE:  THE  BASIS  OF 
DESIGN  IN  NATURE 


"^OR  the  purpose  of  the  present  work,  it  will  be  sufficient  to  deal 
only  with  the  conclusions  obtained  by  the  study  of  the  bases  of 
design  in  nature.  There  are  so  many  fascinating  aspects  of  natural 
form,  so  many  tempting  by-paths,  that  it  would  be  easy  to  wander 
far  from  the  subject  now  under  consideration.  Moreover,  the  mor- 
phological field  has  received  attention  from  many  explorers  more  gifted  and 
better  equipped  to  examine  and  interpret  the  phenomena  of  shape  from  a 
scientific  point  of  view  than  the  writer,  whose  training  has  been,  and  disposi- 
tion is,  merely  that  of  a  practical  artist.4  His  working  hypothesis,  responsible 
for  the  material  here  presented,  was  formulated  upon  the  assumption  that  the 
same  curve  persists  in  vegetable  and  shell  growth.  This  curve  is  known  mathe- 
matically as  the  constant  angle  or  logarithmic  spiral.  This  curiously  fascinating 
curve  has  received  much  attention.5  As  a  curve  form,  its  use  for  purposes  of 
design  is  limited,  but  it  possesses  a  property  by  which  it  may  readily  be  trans- 
formed into  a  rectangular  spiral.  The  spiral  in  nature  is  the  result  of  a  process 
of  continued  proportional  growth.  This  will  be  clear  if  we  consider  a  series  of 
cells  produced  during  a  period  of  time,  the  first  cell  growing  according  to  a  defi- 
nite ratio  as  new  cells  are  added  to  the  system.  (See  Figs,  i  and  2.)  The  shell 
is  but  a  cone  rolled  up.  Fig.  i  represents  the  cone  of  such  an  aggregate,  while 
Fig.  1  shows  the  system  coiled. 


Fig.  i.  Fig.  2. 

The  curve  of  the  coil  is  a  logarithmic  spiral  in  which  the  law  of  proportion  is 
inherent.  A  distinctive  feature  of  this  curve  is  that  when  any  three  radii  vectors 
are  drawn,  equi-angular  distance  apart,  the  middle  one  is  a  mean  proportional 
between  the  other  two;  in  other  words,  the  three  vectors,  or  the  three  lines 
drawn  from  the  center  or  pole  to  the  circumference,  equi-angular  distance  apart, 
form  three  terms  of  a  simple  proportion;  A  is  to  B,  as  B  is  to  C,  and  according 
to  the  "rule  of  three"  the  product  of  the  extremes,  A  and  C,  is  equal  to  the 
square  of  the  mean.  A  multiplied  by  C  equals  B  multiplied  by  itself.  The  early 


I  2 


DYNAMIC  SYMMETRY 


Greeks  covered  the  point  geometrically  when  they  established  the  fact  that 
in  a  right  triangle,  a  line  drawn  perpendicular  to  the  hypotenuse  to  meet  the 
intersection  of  the  legs,  is  the  side  of  a  square  equal  in  area  to  the  rectangle 
formed  by  the  two  segments  of  the  hypotenuse.  (Fig.  3.) 


Fig.  3- 

These  three  lines  C,  B,  A,  constitute  three  terms  in  a  continued  proportion. 

When  the  three  radii  vectors  are  drawn  from  the  center  to  the  circumference 
of  the  shell  curve,  as  in  Fig.  4, 


Fig.  4. 

and  these  points  of  intersection  with  the  spiral  are  connected  by  two  straight 
lines,  a  right  angle  is  created  at  C  and  a  right  triangle  formed,  ACB.  (Fig.  5.) 


Fig-  5- 

If  the  mean  proportional  line  of  this  right  triangle,  ACB,  that  is,  if  the  line 
CO  be  produced  through  the  pole  or  center  of  the  spiral  to  the  opposite  side  of 
the  curve,  obviously  another  right  angle  is  created  as  at  B,  and  by  drawing 
the  line  BD,  the  right  triangle  DBC  is  formed.  (Fig.  6.) 


Fig.  6. 


DYNAMIC  SYMMETRY 


13 


The  process  may  be  extended  until  the  entire  spiral  curve  has  been  trans- 
formed into  a  right  angle  spiral,  as  shown  by  the  lines  AC,  CB,  BD,  DE,  EF, 
etc.,  a  form  suggestive  of  the  Greek  fret.  There  now  exists  in  the  area  bounded 
by  the  spiral  curve  a  double  series  of  lines  in  continued  proportion,  each  line 
bearing  the  same  relation  to  its  predecessor  as  the  one  following  bears  to  it. 

As  far  as  design  is  concerned,  we  may  now  dispense  with  the  curve  of  the 
spiral.  There  have  been  extracted  from  it  all  essentials  for  the  present  purpose 
and  there  remains  but  the  placing  of  the  angular  spiral  within  a  rectangle.  This 
may  be  done  in  any  rectangle  by  drawing  a  diagonal  to  the  rectangle  and  from 
one  of  the  remaining  corners  a  line  to  cut  this  diagonal  at  right  angles.  This 
line,  drawn  from  one  corner  of  the  rectangle  to  cut  the  diagonal  at  right  angles, 
is  produced  to  the  opposite  side  of  the  rectangle.  (Fig.  7.) 


Fig-  7- 

Such  a  line  we  shall  refer  to  as  a  perpendicular,  and  in  all  cases  it  is  drawn  from 
a  corner.  It  establishes  proportion  within  a  rectangle,  and  is  the  diagonal  to  the 
reciprocal  of  the  rectangle.  In  Fig.  8,  AB  is  a  reciprocal  rectangle  and  conse- 
quently is  similar  to  the  rectangle  CD.7 


Fig.  8. 


There  exists  a  series  of  rectangles  whose  sides  are  divided  into  equal  parts 
by  the  perpendicular  to  the  diagonal.  Take  for  example  the  rectangle  in  Fig.  9, 
where  the  line  AB  bisects  the  line  CD,  at  B.  In  such  a  rectangle  a  relationship 
exists  between  the  end  and  the  side  expressed  numerically  by  1,  or  unity,  and 
1.4142  (see  Fig.  10)  or  the  square  root  of  two,  and  a  square  constructed  on  the 
end  is  exactly  one-half,  in  area,  of  the  square  constructed  on  the  side. 


H  DYNAMIC  SYMMETRY 


Fig.  9.  Fig.  10. 


The  student  may  draw  all  the  rectangles  of  Dynamic  Symmetry  with  a 
right  angle  and  a  decimally  divided  scale,  preferably  one  divided  into  milli- 
meters. 

It  will  be  noticed  that  the  number  1.4I42  is  an  indeterminate  fraction.  In 
other  words,  while  the  end  and  the  side  of  this  rectangle  are  incommensurable 
in  line,  they  are  commensurable  in  square.6  This  rectangle  we  may  call  a  root- 
two  rectangle.  It  is  found  to  possess  properties  of  great  importance  to  design. 
It  is  the  rectangle  whose  reciprocal  is  equal  to  half  the  whole.7 


Fig.  11a.  Fig.  n£. 

Fig.  iia  shows  two  perpendiculars  in  the  rectangle,  and  rectangular  spirals 
wrapping  around  two  poles  or  eyes.  If,  as  in  Fig.  n£,  four  perpendiculars  are 
drawn  to  the  two  diagonals,  and  then  lines  at  right  angles  to  the  sides  and  ends 
through  the  intersections,  the  area  of  the  rectangle  will  be  divided  into  similar 
figures  to  the  whole,  the  ratio  of  division  being  two. 


Fig.  12a.  Fig.  12b. 

If,  instead  of  lines  coinciding  with  the  spiral  wrapping,  as  in  Fig.  11a,  lines 
are  drawn  through  the  eyes,  and  at  right  angles  to  the  sides  and  ends,  the  rec- 


DYNAMIC  SYMMETRY  15 

tangle  will  be  divided  into  similar  shapes  to  the  whole,  with  a  ratio  of  three. 
(See  Fig.  12.)  AB  is  one  third  of  AC,  while  AD  is  one  third  of  AE. 

A  rectangle  whose  side  is  divided  into  three  equal  parts  by  horizontal  lines 
drawn  through  the  points  of  intersection  of  the  perpendiculars  and  the  sides  of 
the  rectangle  has  a  ratio  between  its  end  and  its  side  of  1,  or  unity,  to  1.732  or 
the  square  root  of  3.  This  is  a  root-three  rectangle  and  has  characteristics  simi- 
lar to  those  of  a  root-two  rectangle,  except  that  it  divides  itself  into  similar 
shapes  to  the  whole  with  a  ratio  of  3.  AB,  BC  and  CD  are  equal.  (Fig.  13.) 
Lines  drawn  through  the  eyes  of  the  spiral  divide  this  rectangle  into  four  equal 
parts.  The  square  on  the  end  of  this  rectangle  is  one-third  the  area  of  the 
square  on  the  side. 


Fig.  13 

A  rectangle  whose  side  is  divided  into  four  equal  parts  by  a  perpendicular 
has  a  ratio  between  its  end  and  its  side  of  one  to  two,  or  unity  to  the  square 
root  of  four.  This  rectangle  has  properties  similar  to  those  of  a  root-two  or  a 
root-three  rectangle,  except  that  it  divides  itself  into  similar  rectangles  by  a 
ratio  of  four,  and  the  area  of  the  square  on  the  end  is  one-fourth  the  area  of 


X 

\ 

Fig.  14a. 


Fig.  14A 


16  DYNAMIC  SYMMETRY 

the  square  on  the  side.  This  is  a  root-four  rectangle.  Lines  drawn  through  the 
eyes  of  the  spirals  of  a  root-four  rectangle  divide  the  area  into  five  equal  parts 
similar  to  the  whole.  (Fig. 

A  rectangle  whose  side  is  divided  into  five  equal  parts  by  a  perpendicular 
has  a  ratio  between  its  end  and  its  side  of  one  to  "2.236,  or  the  square  root  of 
five.  This  area  is  a  root-five  rectangle  and  it  possesses  properties  similar  to  those 
of  the  other  rectangles  described,  except  that  it  divides  itself  into  rectangles 
similar  to  the  whole  with  ratios  of  five  and  six.  A  square  on  the  end  is  to  a  square 
on  the  side  as  one  is  to  five,  that  is,  the  smaller  square  is  exactly  one-fifth  the 
area  of  the  larger  square.  There  is  an  infinite  succession  of  such  rectangles,  but 
the  Greeks  seldom  employed  a  root  rectangle  higher  than  the  square-root  of 
five. 


Fig.  15a.  Fig.  15^. 


The  root-five  rectangle,  moreover,  possesses  a  curious  and  interesting  prop- 
erty which  intimately  connects  it  with  another  rectangle,  perhaps  the  most  ex- 
traordinary of  all.  To  understand  this  strange  rectangle,  we  must  consider  the 
phenomena  of  leaf  distribution.  This  root-five  rectangle  may  be  regarded  as 
the  base  of  dynamic  symmetry.8 

Closely  linked  with  the  scheme  which  nature  appears  to  use  in  its  construc- 
tion of  form  in  the  plant  world  is  a  curious  system  of  numbers  known  as  a  sum- 
mation series.  It  is  so  called  because  the  succeeding  terms  of  the  system  are 
obtained  by  the  sum  of  two  preceding  terms,  beginning  with  the  lowest  whole 
number;  thus,  1,  2,  3,  5,  8,  13,  21,  34,  55,  89,  144,  etc.  This  conjigigmgjeries  [ 
of  numbers  is  also  known  as  a  Fibonacci  series,  because  it  was  first  noted  by 
Leonardo  da  Pisa,  called  Fibonacci.  Leonardo  was  distinguished  as  an  arith- 
metician and  also  as  the  man  who  introduced  in  Europe  the  Arabic  system  of 


DYNAMIC  SYMMETRY  17 

notation.  Gerard,  a  Flemish  mathematician  of  the  17th  century,  also  drew 
attention  to  this  strange  system  of  numbers  because  of  its  connection  with  a 
celebrated  problem  of  antiquity,  namely,  the  eleventh  proposition  of  the  second 
book  of  Euclid.  Its  relation  to  the  phenomena  of  plant  growth  is  admirably 
brought  out  by  Church,5  who  uses  a  sunflower  head  to  explain  the  phenomena. 

What  is  called  normal  phyllotaxis  or  leaf  distribution  in  plants  is  represented 
or  expressed  by  this  summation  series  of  numbers.  The  sunflower  is  generally 
accepted  as  the  most  convenient  illustration  of  this  law  of  leaf  distribution. 
An  average  head  of  this  flower  possesses  a  phyllotaxis  ratio  of  34  x  55.  These 
numbers  are  two  terms  of  the  converging  summation  series. 

The  present  inquiry  is  concerned  with  only  two  aspects  of  the  phyllotaxis 
phenomena:  the  character  of  the  curve,  and  the  summation  series  of  numbers 
which  represents  the  growth  fact  approximately.9  The  actual  ratio  can  be  ex- 
pressed only  by  an  indeterminate  fraction.  The  plant,  in  the  distribution  of  its 
form  elements,  produces  a  certain  ratio,  1.6 18,  which  is  obtained  by  dividing 
any  one  term  of  the  summation  series  by  its  predecessor.  This  ratio  of  1.618 
is  used  with  unity  to  form  a  rectangle  which  is  divided  by  a  diagonal  and  a 
perpendicular  to  the  diagonal,  as  in  the  root  rectangles.  (Fig.  19.) 


Fig.  19.  Fig.  20. 


"A  fairly  large  head,  5  to  6  inches  in  diameter  in  the  fruiting  condition,  will  show  ex- 
actly 55  long  curves  crossing  89  shorter  ones.  A  head  slightly  smaller,  3  to  5  inches 
across  the  disk,  exactly  34  long  and  55  short;  very  large  1 1  inch  heads  give  89  long  and 
144  short;  the  smallest  tertiary  heads  reduce  to  21  and  34  and  ultimately  13  and  21  may 
be  found;  but  these  being  developed  late  in  the  season  are  frequently  distorted  and  do 
not  set  fruit  well.  A  record  head  grown  at  Oxford  in  1899  measured  22  inches  in  diam- 
eter, and,  though  it  was  not  counted,  there  is  every  reason  to  believe  that  it  belonged 
to  a  still  higher  series  (144  and  233). 

"Under  normal  conditions  of  growth  the  ratio  of  the  curves  is  practically  constant. 
Out  of  140  plants  counted  by  Weisse,  6  only  were  anomalous,  the  error  thus  being  only 
4  per  cent."  A.  H.  Church,  "On  the  Relation  of  Phyllotaxis  to  Mechanical  Law."5 


l8 


DYNAMIC  SYMMETRY 


Thus,  we  may  call  this  "the  rectangle  of  the  whirling  squares/'  because  its 
continued  reciprocals  cut  off  squares.  The  line  AB  in  Fig.  19  is  a  perpendicular 
cutting  the  diagonal  at  a  right  angle  at  the  point  O,  and  BD  is  the  square  so 
created.  BC  is  the  line  which  creates  a  similar  figure  to  the  whole.  One  or  unity 
should  be  considered  as  meaning  a  square.  The  number  2  means  two  squares, 
3,  three  squares,  and  so  on.  In  Fig.  19  we  have  the  defined  square  BD,  which 
is  unity.  The  fraction  .618  represents  a  shape  similar  to  the  original,  or  is  its 
reciprocal.  Fig.  20  shows  the  reason  for  the  name  "rectangle  of  the  whirling 
squares."  1,2,3, 4, 5, 6,  etc.,  are  the  squares  whirling  around  the  pole  O. 


.{>IB  ^ 

Fig.  21. 

If  the  ratio  1.6 1 8  is  subtracted  from  2.236,  the  square  root  of  5,  the  remainder 
will  be  the  decimal  fraction  .618.  This  shows  that  the  area  of  a  root-five  rec- 
tangle is  equal  to  the  area  of  a  whirling  square  rectangle  plus  its  reciprocal, 
that  is,  it  equals  the  area  of  a  whirling  square  rectangle  horizontal  plus  one 
perpendicular,  as  in  Fig.  21. 

The  writer  believes  that  the  rectangles  above  described  form  the  basis  of 
Egyptian  and  Greek  design.  In  the  succeeding  chapters  will  be  explained  the 
technique  or  method  of  employment  of  these  rectangles  and  their  application 
to  specific  examples  of  design  analysis. 


CHAPTER  TWO:  THE  ROOT 
RECTANGLES 


r 


"^HE  determination  of  the  root  rectangles  seems  to  have  been 
one  of  the  earliest  accomplishments  of  Greek  geometers.9  In 
fact,  geometry  did  not  become  a  science  until  developed  by 
the  Greeks  from  the  Egyptian  method  of  planning  and  sur- 
veying. The  development  of  the  two  branches  of  the  same 
idea  went  together.  Greek  artists,  working  upon  this  basis  to  elaborate  and 
perfect  a  scheme  of  design,  labored  side  by  side  with  Greek  philosophers,  who 
examined  the  idea  to  the  end  that  its  basic  principles  might  be  understood  and 
applied  to  the  solution  of  problems  of  science.  How  well  this  work  was  done, 
Greek  art  and  Greek  geometry  testify. 

As  early  as  the  Sixth  Century  B.  C.  Greek  geometers  were  able  to  "deter- 
mine a  square  which  would  be  any  multiple  of  a  square  on  a  linear  unit."  It  is 
evident  that  in  order  to  construct  such  squares  the  root  rectangle  must  be  em- 
ployed. We  find  the  Greek  point  of  view  essentially  different  from  ours,  in  con- 
sidering areas  of  all  kinds.  We  regard  a  rectangular  area  as  a  space  inclosed  by 
lines,  and  the  ends  and  sides  of  the  majority  of  root  rectangles,  because  these 
lines  are  incommensurable,  would  now  be  called  irrational.  The  Greeks,  how- 
ever, put  them  in  the  rational  class,  because  these  lines  are  commensurable  in 
square.6  This  conception  leads  directly  to  another  Greek  viewpoint  which 
resulted  in  the  evolution  of  a  method  employed  by  them  for  the  solution  of 
geometric  problems,  to  wit,  "the  application  of  areas."10  Analysis  of  Greek 
design  shows  a  similar  idea  was  used  in  art  when  rectangular  areas  were 
exhausted  by  the  application  of  other  areas,  for  example,  the  exhaustion  of  a 
rectangle  by  the  application  of  the  squares  on  the  end  and  the  side,  in  order 
that  the  area  receiving  the  application  might  be  clearly  understood  and  its  pro- 
portional parts  used  as  elements  of  design.  If  the  square  on  the  end  of  a  root- 
two  rectangle  be  applied  to  the  area  of  the  rectangle,  it  "falls  short,"  is  "elliptic," 
and  the  part  left  over  is  composed  of  a  square  and  a  root-two  rectangle.  (See 
Fig.  la.)  If  the  same  square  be  applied  to  the  other  end,  so  as  to  overlap  the 
first  applied  square,  the  area  of  the  rectangle  is  divided  into  three  squares  and 
three  root-two  rectangles.  (See  Fig.  lb.)  And,  if  the  square  on  the  side  of  a  root- 
two  rectangle  be  applied,  it  "exceeds,"  is  "hyperbolic,"  and  the  excess  is  com- 
posed of  two  squares  and  one  root-two  rectangle.11  (See  Fig.  ic.) 

This  idea  is  quite  unknown  to  modern  art,  but  that  it  is  of  the  utmost  im- 
portance will  be  shown  in  this  book  by  the  analyses  of  the  Greek  vases. 

Let  us  now  consider  various  methods  of  construction  of  the  root  rectangles, 


20 


DYNAMIC  SYMMETRY 


and,  of  course,  the  whirling  square  rectangle.  We  will  commence  with  the  latter, 
which  is  intimately  connected  with  extreme  and  mean  ratio,  a  geometrical  con- 
ception of  great  artistic  and  scientific  interest  to  the  early  Greeks.  Using  dy- 
namic symmetry,  this  problem  of  cutting  a  line  in  extreme  and  mean  ratio  may 
be  solved  through  subtracting  unity  from  the  diagonal  of  a  root-four  rectangle: 
the  Greek  method  was  not  essentially  different.  To  the  early  geometers  it  was 
the  cutting  of  a  line  so  that  the  rectangle  formed  by  the  whole  line  and  the  lesser 
segment  would  equal  the  area  of  the  square  described  on  the  greater  segment.5 


J" 

S 

5 

f2 

S 

7^2 

S 

Fig.  \a. 


Fig.  \b. 


Fig.  ic. 


Euclidean  construction  furnishes  an  easy  method  for  describing  not  only 
the  whirling  square,  but  also  the  root-five  rectangle,  after  the  following  man- 
ner: A  square  is  drawn  and  one  side  bisected  at  A.  The  line  AB  is  used  as  a 
radius  and  the  semi-circle  CBFD  described.  DE  is  a  root-five  rectangle.  BC 
and  DF  are  rectangles  of  the  whirling  square,  as  are  also  CF  and  BD.  (Fig.  2.) 


/ 

^  X 

N 

/ 

/ 

\ 

/ 

\ 

\ 

/ 

/ 

\ 

\ 

/ 

\ 

/ 

/ 

/ 

/ 

\ 

\ 

\ 

Fig.  1. 

The  relation  of  the  rectangles,  which  have  been  described,  to  certain  com- 
pound shapes  derived  from  them  will  now  be  shown.  If,  in  a  rectangle  of  the 
whirling  squares  mapped  out  as  in  Fig.  3,  a  !ine  parallel  to  the  sides  be  drawn 
through  the  eyes  A  and  B,  it  cuts  from  the  major  shape  a  root-five  rectangle, 
i.  <?.,  a  square  and  two  whirling  square  rectangles,  C,  D,  and  E, — D  being  the 
square.  Fig.  4  shows  how  a  line  drawn  through  the  eyes  F  and  G,  parallel  to 


DYNAMIC  SYMMETRY 


21 


the  end,  defines  also  a  root-five  rectangle,  C  being  the  square.  Obviously  this 
may  be  done  at  either  end  and  side,  resulting  in  the  determination  of  four 
root-five  rectangles  overlapping  each  other  within  the  major  shape.  In  a  whirl- 
ing square  rectangle  (Fig.  5*3),  if  lines  be  drawn  through  the  eyes  A,  B,  C,  D 
parallel  to  the  ends,  and  A  and  B  connected  by  another  line,  an  area  will  be 


c'''  I 

Fig-  3-  Fig.  4. 

defined,  composed  of  the  square  E  and  the  rectangle  F.  This  shape,  composed 
of  E  and  F,  is  numerically  described  as  the  rectangle  1.382.  The  square  E  is 
unity.  The  rectangle  F  is  the  fraction  .382,  this  being  the  reciprocal  of  2.618, 
i.  e.,  it  is  a  whirling  square  rectangle,  1.618  plus  1.  (Fig.  $b.)  If  this  1.382 
rectangle  is  divided  by  2,  the  shapes  G,  H  (Fig.  5c),  result  and  each  is  composed 
of  a  square  and  a  root-five  rectangle.  1.382  divided  by  2  equals  .691,  which, 
divided  into  unity,  proves  to  be  the  reciprocal  of  1.4472,  and  .4472  is  the  recip- 
rocal of  root-five  and  is  itself  a  root-five  rectangle.  Many  Greek  vases  were 
constructed  according  to  the  principles  inherent  in  this  1.382  shape. 


Fig.  5a. 


Fig.  $b. 


Fig.  $c 


If  a  whirling  square  rectangle  is  subtracted  from,  or  applied  to,  a  square,  the 
defect  is  .382  or  a  whirling  square  rectangle  plus  a  square.  (See  Fig.  6.)  .618 
subtracted  from  1.  equals  .382.  If,  as  in  Fig.  7,  a  whirling  square  rectangle  is 


22 


DYNAMIC  SYMMETRY 


placed  in  the  center  of  the  shape  1.38a,  the  "defect"  area  on  either  side  is  com- 
posed of  a  square  and  a  whirling  square  rectangle. 


Fig.  7- 


The  reciprocal  of  1.382  is  .7236;  .4472  multiplied  by  2  equals  .8944,  and 
this  result  added  to  .7236  equals  1.618.  (See  Fig.  8.)  The  area  of  Fig.  8  is  com- 
posed of  two  root-five  rectangles,  .4472  x  2,  plus  a  .7236  shape. 


Fig.  8. 

All  of  these  shapes  are  found  in  abundance  in  both  Egyptian  and  Greek  art. 

The  square  is  considered  the  unit  form  or  monad.  "Iamblicus  (fl.  circa  300 
A.  D.)  tells  us  that  .  .  .  'an  unit  is  the  boundary  between  number  and 
parts  because  from  it,  as  from  a  seed  and  eternal  root,  ratios  increase  recip- 
rocally on  either  side,'  i.  e.,  on  one  side  we  have  multiple  ratios  continually 
increasing,  and  on  the  other  (if  the  unit  be  subdivided),  submultiple  ratios  with 
denominators  continually  increasing."  ("The  Thirteen  Books  of  Euclid's  Ele- 
ments," by  T.  F.  Heath,  Def.  Book  VII.) 

The  Reciprocal  Ratios  Within  a  Square 
The  root  rectangles  are  constructed  within  a  square  by  the  simple  geometri- 
cal method  shown  in  Fig.  9.  AB  is  a  quadrant  arc  with  center  D  and  radius  DB. 
DC  is  a  diagonal  to  a  square  and  it  cuts  the  quadrant  arc  at  F.  A  line,  parallel 
to  a  side  of  the  square,  is  drawn  through  F.  This  line  determines  a  root-two 


DYNAMIC  SYMMETRY  23 

rectangle  and  DE  is  its  diagonal.  A  diagonal  to  a  root-two  rectangle  cuts  the 
quadrant  arc  at  H.  GD  is  a  root-three  rectangle,  the  diagonal  of  which  cuts  the 
quadrant  arc  at  J.  DI  is  a  root-four  rectangle  and  its  diagonal  cuts  the  quad- 
rant arc  at  L.  DK  is  a  root-five  rectangle  and  so  on.  All  the  root  rectangles  may 
be  thus  obtained  within  a  square. 


A 

^               /  - 
N  / 

V/  a 

/  \  / 
/          \  / 

/              \»/  0 

/          /  \ 
/          /  \ 

/       /  /  \ 
/      /  /  X 

s 

Fig.  9. 


The  root  ratios  outside  of  a  square  are  obtained  from  diagonals,  Fig.  10. 

AB,  the  diagonal  of  the  unit  form  or  square,  determines  the  point  C,  the  side 
of  a  root-two  rectangle.  The  diagonal  of  a  root-two  rectangle,  as  AD,  becomes 
the  side  of  a  root-three  rectangle,  as  AE.  AF,  the  diagonal  of  a  root-three  rec- 
tangle, becomes  the  side  of  a  root-four  rectangle,  as  AG.  AH,  the  diagonal  of  a 
root-four  rectangle,  becomes  the  side  of  a  root-five  rectangle,  as  AI.  AJ,  the 
diagonal  of  a  root-five  rectangle  becomes  the  side  of  a  root-six  rectangle,  and 
so  on  to  infinity.  In  any  of  these  rectangles  a  square  on  the  end  is  some  even 
multiple  of  a  square  on  the  side.  The  square  constructed  on  the  line  AC  is  dou- 
ble the  square  on  AK;  the  square  on  AE  is  three  times  the  area  of  the  square  on 
AK;  the  square  on  AG  is  four  times  the  square  on  AK;  the  square  on  AI  is  five 
times  the  square  on  AK,  etc.  This  was  the  Greek  method  of  describing  squares 
which  would  be  any  multiple  of  a  square  on  a  given  linear  unit.5  The  given  linear 
unit  is  the  line  AK.  The  rectangles  inside  the  square  are  the  reciprocals  of  the 
rectangles  outside  the  square.  A  root-two  rectangle  inside  the  square,  for  ex- 
ample, is  one-half  the  area  of  the  root-two  rectangle  outside  the  same  square; 
a  root-three  inside,  one-third  of  a  root-three  outside;  a  root-four  inside,  one- 


24 


DYNAMIC  SYMMETRY 


fourth  of  a  root-four  outside  and  a  root-five  inside,  one-fifth  of  a  root-five  out- 
side. And  a  reciprocal  to  any  rectangle  is  obtained  by  drawing  a  perpendicular 
from  one  corner. 


Fig.  10. 


The  whirling  square  rectangle  and  the  root-five  rectangle  are  placed  within 
a  square  thus: 


Fig.  i ia. 


Fig.  i \b. 


The  square  is  first  bisected  by  the  line  AB,  to  obtain  a  root-four  rectangle 
or  two  squares.  From  the  diagonal  of  this  rectangle  CB,  unity,  or  BE,  is 
subtracted  to  determine  the  point  D,  and  CD,  furnishes  the  side  of  the  whirl- 
ing square  rectangle  FE.  See  Fig.  \\a.  A  line  drawn  through  the  point  D, 
parallel  to  a  side  of  the  square,  determines  the  root-five  rectangle  GH.  Fig. 
lib. 

In  a  whirling  square  rectangle  inscribed  in  a  square,  if  lines  be  drawn  through 
the  eyes  and  produced  to  the  opposite  side  of  the  square,  a  root-five  rectangle  is 


DYNAMIC  SYMMETRY  25 

constructed  in  the  center  of  the  square,  see  Fig.  11a.  The  area  AB  is  this  area, 
and  if  these  lines  be  made  to  terminate  at  their  intersection  with  the  diagonals 
of  the  square,  the  whirling  square  rectangle  CD,  is  denned  as  in  Figs.  \ib 
and  1  ic.  That  this  construction  was  used  by  the  Egyptians  in  design  is  shown 
by  the  bas-relief  in  the  form  of  a  square  herewith  reproduced: 


-  J. 


_9/ 


\H\/UV 


-A 


>/  S 


1--- 


Fig.  11a. 


Fig.  11b. 


Fig.  lie. 


When,  as  in  Fig.  13,  a  whirling  square  rectangle  is  comprehended  within  a 
square,  CD,  the  small  square,  AB,  has  a  common  center  with  the  large  square, 
CK,  and  if  the  sides  of  this  small  square,  AB,  are  produced  to  the  sides  of  the 
large  square,  CK,  four  whirling  square  rectangles,  overlapping  each  other  to 
the  extent  of  the  small  square,  AB,  are  comprehended  in  the  major  square. 
They  are  HK,  EF,  CD,  and  CJ,  and  the  major  square  becomes  a  nest  of 
squares  and  whirling  square  rectangles. 


Fig.  13. 


Analysis  of  the  Egyptian  bas-relief  composition  (Fig.  14)  shows  that  its 
designer  not  only  proportioned  the  picture  but  also  the  groups  of  hieroglyphs 
by  the  application  of  whirling  square  rectangles  to  a  square.  The  outlines  of 


26 


DYNAMIC  SYMMETRY 


the  major  square  are  carefully  incised  in  the  stone  by  four  bars,  two  of  which 
have  slight  pointed  projections  on  either  end.  The  general  construction  was 
that  of  a  in  Fig.  12.  Spacing  for  additional  elements  of  the  design  is  shown 
in  c,  Fig.  12,  while  b,  Fig.  12,  exhibits  the  grouping  of  the  hieroglyphic  writing. 


Fig.  14. 


Another  bas-relief  from  Egypt  shows  also  how  a  square  which  is  denned  by 
bars  cut  in  the  stone  at  the  top  and  bottom  of  the  composition  has  its  area 
dynamically  divided  for  a  pictorial  composition.  In  this  example  the  designer 
has  used  a  root-five  rectangle  in  the  center  of  a  square,  Fig.  12a.  The  plan  of 
this  arrangement  is  obvious,  Fig.  15. 

A  simple  theme  in  root-two  is  exhibited  in  Fig.  16.  A  goddess  is  pictured 
supporting  a  formalized  sky  in  the  shape  of  a  bar.  The  spaces  between  the  bars 
on  either  side  of  the  figure  were  filled  with  hieroglyphic  writing.  These  have 
been  omitted  in  this  reproduction.  The  overall  shape  of  this  composition  is  a 


DYNAMIC  SYMMETRY 


28 


DYNAMIC  SYMMETRY 


root-two  rectangle  and  the  simple  method  of  construction  is  shown  in  Fig.  17. 
BC  is  a  square  and  the  side  of  the  rectangle  is  equal  in  length  to  the  diagonal 
of  this  square: 


Fig.  17. 


AB  equals  BC.  DB  and  EF  are  root-two  rectangles,  the  side  of  each  being  equal 
to  half  the  diagonal  of  the  major  square,  or  the  line  BG.  Diagonals  to  the  whole 
intersect  the  side  of  the  major  square  at  the  points  D  F. 

Another  theme  in  root-two  is  disclosed  in  Fig.  18.  The  general  shape  is  a 
square,  carefully  defined  by  incised  lines,  as  in  the  other  examples. 


Fig.  18. 


DYNAMIC  SYMMETRY 


29 


\ 

 ^ 

Q  B 

/ 

V 

F 

v: 

Z  \ 

e 

ITS. 

y~2 

\ 

i. 

H 

'.  * 
P  £ 

&  

K 

Fig.  19a. 


Fig.  19^. 


The  plan  scheme  of  this  design  is  shown  in  Fig.  19*3.  AB,  CD,  AE  and  FG, 
are  four  root-two  rectangles  overlapping  each  other  in  the  major  square,  and  the 
side  of  each,  as  CG,  is  equal  to  half  the  diagonal. of  the  major  shape.  These 
rectangles  subdivide  the  area  of  the  major  square  into  five  squares  and  four 
root-two  rectangles.  In  Fig.  19^,  the  use  of  this  spacing,  in  its  direct  applica- 
tion to  the  design,  is  shown.  The  central  portion  of  the  major  square,  composed 
of  the  square  HG  and  the  root-two  rectangle  HL,  is  divided  by  the  diagonals 
and  perpendiculars  of  this  rectangle.  B  is  the  center  of  the  semicircle  and  BC 
is  made  equal  to  BA.  This  fixes  the  proportion  of  space  to  be  occupied  by  the 
hawk  and  the  field  of  formalized  lotus  flowers.  MJ  is  composed  of  the  two 
squares  MD,  DI  and  the  root-two  rectangle  IJ.  The  square  MD  is  divided  into 
three  parts  and  one  of  these  parts  forms  the  platform  on  which  stands  the 
hippopotamus  god.  This  god  is  placed  within  the  space  KI.  The  same  con- 
struction applies  to  the  other  side  of  the  composition. 

The  examples  of  Egyptian  bas-relief  compositions  described  are,  with  one 
exception,  arrangements  within  a  square.  These  are  used  because  of  their 
obvious  character.  Like  Greek  temples  and  vase  designs,  the  best  Egyptian 
bas-relief  plans  are  composed  within  the  figures  of  dynamic  symmetry,  both 
simple  and  compound. 

The  Egyptians  were  regarded  by  the  Greeks  as  masters  of  figure  dissection. 
The  rational  combinations  of  form,  which  we  may  recover  from  their  designs, 
confirms  this  and  sheds  some  light  on  the  significance  of  the  ceremonial  when 
"the  king,  with  the  golden  hammer,"  drove  the  pins  at  the  points  established 
by  the  harpedonaptae,  the  surveyors  or  "rope-stretchers,"  who  "corded  the 
temple"  and  related  the  four  corners  of  the  building  with  the  four  corners  of  the 
universe.2 


CHAPTER  THREE:  THE  LEAF 


"^HE  rectangles  of  dynamic  symmetry  consist  of  the  root  rec- 
tangles, the  rectangle  of  the  whirling  squares,  and  compound 
shapes  derived  from  subdivision  or  multiplication  of  either 
the  square  root  forms  or  the  rectangle  of  the  whirling 
squares. 

In  both  Greek  and  Egyptian  design  the  compound  shapes  derived  from  the 
rectangle  of  the  whirling  squares  and  the  root-five  shape  greatly  preponderate. 
The  rectangle  of  the  whirling  squares,  as  a  separate  design  shape,  appears, 
but  seldom.  This  fact  suggests  that  extreme  and  mean  ratio,  per  se,  has  little 
aesthetic  significance.  Its  chief  feature  appears  to  be  its  power  as  a  coordinating 
factor  when  used  with  certain  of  the  compound  rectangles. 

There  is  unquestionable  documentary  evidence  that  the  use  of  the  compound 
rectangles,  found  so  plentifully  in  Greek  art,  was  not  arbitrary.  Their  bases 
exist  in  nature  and  it  is  historical  that  the  Greeks  thoroughly  understood  the 
source  from  which  they  are  derived.  (See  the  Thirteenth  Book  of  Euclid's 
Elements.)  Their  discovery  in  nature  by  the  writer  resulted  from  examination 
of  the  trussing  of  a  maple  leaf.  The  shape  of  this  leaf  strikingly  resembles  a 
regular  pentagon. 


Fig.  la. 


Fig.  ib. 


The  leaf  is  shown  above  in  Fig.  la,  and  the  resemblance  of  the  shape  itself 
and  of  its  trussing  to  the  regular  pentagon  and  its  diagonals,  is  apparent  in 
Fig.  ib.  In  a  regular  pentagon  inscribed  in  a  circle  the  relation  of  the  radius 
of  the  escribed  circle  to  the  radius  of  the  inscribed  circle  is  i :  .809.  The  fraction 
.809  multiplied  by  2  equals  1.6 18,  or  the  ratio  of  the  whirling  square  rectangle. 
This  means  that  if  we  escribe  a  square  to  the  circle  escribing  a  regular  penta- 
gon (Fig.  2),  the  area  shown  by  the  heavy  lines  is  represented  by  the  ratio 
1.809.  A  is  a  square  and  B  two  whirling  square  rectangles.  This  is  a  ratio  often 
found  in  Greek  design,  among  amphorae  and  skyphoi  especially.  The  division 
of  the  pentagon  with  its  escribed  square  produces  two  such  areas,  as  in  Fig.  3. 


DYNAMIC  SYMMETRY 


31 


Fig.  1.  Fig.  3. 


In  Fig.  4,  the  point  B  in  reference  to  the  center  A,  is  eighteen  degrees  and  the 
natural  sine  of  eighteen  degrees  or  the  line  AC,  is  .309.  This  fraction  multiplied 
by  1  equals  .618.  The  rectangle  AB,  therefore,  is  composed  of  two  whirling 
square  rectangles,  placed  end  to  end,  a  common  shape  in  Greek  design.  The 
entire  area  shown  by  the  heavy  lines  in  Fig.  5,  is  composed  of  four  whirling 
square  rectangles,  two  perpendicular  side  by  side,  and  two  horizontal  end  to 
end. 


Fig.  4.  Fig.  5. 


A  root-five  rectangle  is  composed  of  a  whirling  square  rectangle,  plus  its 
reciprocal,  or  1.618  plus  .618.  Consequently  the  area  shown  by  the  heavy  lines 
in  Fig.  6a  is  composed  of  two  root-five  rectangles,  and  the  area  in  b,  defined  by 
heavy  lines,  is  equal  to  four  root-five  rectangles. 


Fig.  6a.  Fig.  6b. 


32  DYNAMIC  SYMMETRY 

The  total  distance  AB  in  Fig.  7,  is  1.809.  BC  is  .809,  CD  is  .309,  AC  is  1  or 
unity,  and  AD  is  unity  minus  .309,  or  .691.  This  fraction  .691,  is  the  reciprocal 
of  1.4472,  or  a  square  plus  a  root-five  rectangle.  ED  is  this  shape,  the  key  to  the 
Parthenon  plan  and  many  other  Greek  designs.  It  is  a  favorite  shape  for  many 
vases. 12  „   


Fig.  7. 

The  intersection  of  two  diagonals  to  the  pentagon,  in  Fig.  8,  determines 
the  area  shown  by  the  heavy  lines,  which  is  composed  of  two  squares  and 
two  root-five  rectangles  or  the  ratio  1.382. 


Fig.  8. 


The  distance  AB  in  Fig.  9,  is  the  difference  between  1.809  and  2,  or  .191,  and 
this  fraction  multiplied  by  2  equals  .382,  the  reciprocal  of  2.618.  Therefore 
the  area  AD  is  composed  of  four  shapes,  two  squares  and  two  whirling  square 
rectangles. 


Fig.  9. 


DYNAMIC  SYMMETRY  .  33 

The  radius  of  a  circle  escribing  a  pentagon  is  1,  and  the  radius  of  the  inscribed 
circle  is  .809.  Therefore  the  area  AB,  in  Fig.  10,  is  composed  of  two  whirling 
square  rectangles.  The  area  BC  plus  AD  is  composed  of  eight  squares  and 


Fig.  10. 


eight  whirling  square  rectangles.  If  these  areas  BC,  AD,  are  placed  one  over 
the  other,  the  area  is  then  expressed  as  5.236,  i.  e.,  1.236  plus  four  squares. 
The  reciprocal  of  5.236  is  .191.  (Fig.  11.) 


-t- 

+ 

4 

Fig.  11. 
The  area  5.236. 


The  relation  of  the  diameter  of  the  inscribed  circle  of  a  pentagon  to  the  diam- 
eter of  the  escribed  circle  is  the  ratio  1.236,  i.  e.,  root  five,  2.236,  minus  1,  or 
.618  multiplied  by  2  (the  reciprocal  of  1.236  is  .809).  When  the  squares  escribing 
these  circles  are  placed  in  position,  it  will  be  apparent  that  the  larger  square 
is  greater  than  the  smaller  square  by  sixteen  whirling  square  rectangles  and 
twelve  squares.  (Fig.  12.) 


Fig.  12.' 

When  four  squares  are  placed  in  the  pentagonal  construction,  as  AB  in  Fig. 
13,  the  area  shown  by  the  heavy  lines  is  composed  of  two  rectangles,  each  of 


34  DYNAMIC  SYMMETRY 

which  consists  of  a  square  and  two  whirling  square  rectangles  or  the  ratio 
1.309. 


/  / 

1.309 1 

Fig.  13. 

The  area  AB,  in  Fig.  14,  is  composed  of  a  square  and  a  root-five  rectangle,  as 
is  also  the  area  BC.  The  areas  BD,  BE,  are  1.309  rectangles. 


Fig.  14. 


The  ratio  1.382  is  obtained  by  dividing  1.309  into  1.809.  It  is  represented  by 
the  area  AB  in  Fig.  15,  and  consists  of  a  square  plus  .382  and  this  fraction  is  the 
reciprocal  of  2.618,  i.  e.,  a  square  plus  a  whirling  square  rectangle.  Also,  if  this 
ratio  of  1.382  is  divided  by  two,  it  will  be  noticed  that  the  area  could  be  expressed 
by  two  .691  shapes,  each  of  which  is  the  reciprocal  of  1.4472  or  a  square  plus  a 
root-five  rectangle.  The  area  BC  is  a  whirling  square  rectangle,  .691  divided 
into  1.1 18  producing  1.618.  The  area  CD  is  a  square. 


DYNAMIC  SYMMETRY  35 

A  line  drawn  through  the  intersection  of  two  diagonals  of  a  pentagon  divides 
the  area  of  the  major  square  as  in  Fig.  16,  into  three  shapes,  two  of  which  are 
rectangles  of  the  whirling  squares  and  one  is  composed  of  a  square  and  a  root- 
five  rectangle. 


Fig.  16. 


When  the  area  of  a  major  square  is  subdivided,  as  in  Fig.  17,  four  very  in- 
teresting shapes  result.  AB  is  a  rectangle  of  the  whirling  squares.  BC  is  rep- 
resented by  the  ratio  1.1708,  this  being  composed  of  .618  plus  .5528,  the  latter 
ratio  being  the  reciprocal  of  1.809  or  a  square  plus  two  whirling  square  rec- 
tangles. The  ratio  1.1708  could  be  expressed  by  .4472  plus  .7236.  The  rectangle 
BD  is  the  ratio  1.7236,  a  square  plus  .7236,  this  fraction  being  the  reciprocal 
of  1.382.  The  area  BE,  representing  the  ratio  1.099,  *s  a  complicated  but  very 
important  shape.  That  it  was  used  by  the  Greeks  with  telling  effect  is  evi- 
denced by  a  bronze  wine  container  of  the  Fifth  Century  B.  C,  now  in  the 
Museum  of  Fine  Arts  in  Boston. 


D  /-/?/  x  .809 


8 

A 

Fig.  17. 


Two  of  the  four  rectangles  in  Fig.  18  have  been  described.  The  area  BD, 
being  1.0652,  consists  of  a  whirling  square  rectangle  plus  a  root-five  rectangle, 
.618  plus  .4472.  BE  is  a  1.382  rectangle. 


36 


DYNAMIC  SYMMETRY 


Fig.  1 8.  Fig.  19. 


The  area  of  the  major  square  in  Fig.  19  is  divided  into  twelve  shapes.  Two 
are  squares.  AB  consists  of  two  .382  or  2.618  rectangles,  CD  four  such  figures, 
while  BC  consists  of  four  .854  shapes.  This  .854  shape  is  valuable.  It  consists 
of  .618  plus  .236;  the  latter  being  the  reciprocal  of  4.236  or  root-five  plus  two. 
The  ratio  .854  is  the  reciprocal  of  1.1708. 

Seven  of  the  thirteen  subdivisional  figures  in  Fig.  20  are  squares.'  AB  is  a 
square  and  BC  consists  of  a  square  plus  two  root-five  rectangles,  the  ratio 
being  1.8944,  and  its  reciprocal  .528.  The  area  BD  is  represented  by  the  ratio 
2. 1 1 8,  i.  <?.,  root  five,  2.236  divided  by  two,  1.118,  plus  one. 


Is  rs 

s 

-rs  \\ 

rs  / 

rs 

D, 

Fig.  20. 

The  rectangle  AB  in  Fig.  21,  has  a  ratio  of  1.4472;  a  square  plus  a  root-five 
rectangle,  .4472  being  one-fifth  of  2.236  and  a  reciprocal  of  that  number. 


DYNAMIC  SYMMETRY  37 

The  area  AB  in  Fig.  11a  is  a  whirling  square  rectangle,  1.809  on  tne  s^e  and 
1.1 18  on  the  end.  CB  is  the  major  square  of  this  rectangle.  The  shape  DE  is 
the  ratio  1.2764,  i.  e.,  .691  divided  into  .882.  Of  this  area  .691  by  .691  makes  a 
square,  and  .191,  the  difference  between  .691  and  .882,  divided  into  .691  fur- 
nishes 3.618,  i.  e.,  a  whirling  square  rectangle  plus  two  squares.  The  area  BD, 
.882  by  1.1 18,  supplies  the  ratio  1.267.  This  ratio  is  more  easily  recognized  if 
we  consider  its  reciprocal  .7888.  Four  root-five  rectangle  reciprocals  equal  the 
ratio  1.7888,  .4472  multiplied  by  four.  .7888,  therefore,  is  four  root-five  rectan- 
gles minus  one. 

It  is  a  beautiful  shape  and  may  be  obtained  readily  from  the  whirling  square 
rectangle.  This  particular  ratio  was  discovered  independently  by  Wm.  Sergeant 
Kendall,  in  the  form  of  overlapping  whirling  square  rectangles  creating  a  root- 
five  rectangle  by  their  union  as  in  Fig.  11b. 


Fig.  22a.  Fig.  21b. 

The  area  AB  in  Fig.  23  is  composed  of  two  squares  and  two  root-five  rec- 
tangles, or  the  ratio  2.8944,  i.  e.,  1.4472  multiplied  by  two;  .691  divided  into 
2.000.  The  fraction  is  not  quite  .691,  but  this  number  is  sufficiently  close  for  all 
practical  purposes.  BC  and  CD  are  two  equal  areas  each  composed  of  a  square 
and  two  whirling  square  rectangles,  i.  e.,  each  has  a  ratio  of  1.309. 


Fig.  23. 


38  DYNAMIC  SYMMETRY 

In  Fig.  24  the  area  AB,  unity  on  the  end  and  1.191  on  the  side,  is  a  square 
plus  .191  and  this  fraction  represents  two  squares  and  two  whirling  square 
rectangles.  The  area  BC  represents  four  whirling  square  rectangles;  .618  mul- 
tiplied by  four,  2.472  minus  one  or  1.472.  DE  is  a  root-five  rectangle.* 


Fig.  24. 


The  area  AB  in  Fig.  25  is  divided  into  squares,  root-five  rectangles  and  rec- 
tangles of  the  whirling  squares. 

In  Fig.  26  the  area  AB  is  composed  of  two  rectangles  each  consisting  of  a 
square  plus  .382,  this  fraction  being  the  reciprocal  of  2.618.  The  area  AB  may 
be  expressed  also,  as  a  square  and  a  root-five  rectangle,  1.4472.  The  area  BC 
is  composed  of  two  whirling  square  rectangles. 


The  whirling  square  rectangle  AB  in  Fig.  27  may  also  be  expressed  as  two 
squares  and  two  root-five  rectangles. 

The  area  AB  in  Fig.  28,  consists  of  six  whirling  square  rectangles.  The  side 
of  this  rectangle  is  2.000  and  the  end  .927. 

*  Euclid,  XIII,  I,  in  substance  proves  that  a  rectangle  which  is  .809  on  the  end  and 
1.809  on  the  side  is  a  root-five  rectangle. 


DYNAMIC  SYMMETRY 


39 


y 

Fig.  27. 


s 

rs 

rs 

ws 

Fig.  28. 


In  the  thirteenth  book  of  the  Elements,  Euclid  proves  the  relationship  of 
the  end,  side  and  diagonal  of  the  whirling  square  rectangle.  Proposition  8  is 
devoted  to  proving  that  diagonals  to  a  pentagon  cut  each  other  in  the  propor- 
tion of  the  whirling  square  rectangle.  The  fact  enunciated  in  this  proposition 
suggests  the  reason  why  the  Pythagoreans  of  the  Sixth  Century  B.  C.  used 
the  Pentagram  as  a  symbol  of  their  school. 


The  first  six  propositions  of  the  13th  book  are  devoted  to  the  consideration  of 
the  relationships  of  areas  described  on  lines  connected  with  the  whirling  square 
rectangle.  In  the  first  proposition  the  geometrical  construction  brings  out  the 
fact  that  a  rectangle,  the  end  of  which  is  .809  and  the  side  1.809,  is  a  root-five 
rectangle.  In  the  9th  proposition  proof  is  furnished  that  the  side  of  a  hexagon 
and  the  side  of  a  decagon  added,  form  a  line  which  is  cut  in  extreme  and  mean 
ratio,  and  the  side  of  the  hexagon  is  the  greater  segment.  (Fig.  31.)  Proposi- 
tion 10  furnishes  the  proof  that  the  square  on  the  side  of  a  pentagon  inscribed 
in  a  circle  is  equal  in  area  to  the  squares  on  the  sides  of  a  hexagon  and  a  decagon 
inscribed  in  the  same  circle.  Fig.  31*2  shows  this  relationship.  This  figure  is  of 
necessity  a  right-angled  triangle. 


4-o  DYNAMIC  SYMMETRY 


Fig.  31. 


Fig.  31a. 


Later,  in  XIII,  18,  the  rectangular  relationship  is  more  clearly  shown  in  a 
root-five  rectangle.  The  Euclidean  diagram  of  the  1 8th  proposition  is  peculiarly 
interesting  in  the  light  of  dynamic  symmetry  because  it  suggests  what  may 
have  been  the  Greek  method  of  constructing  the  dynamic  rectangles  in  a 
square. 

The  writer's  method  of  describing  a  root-five  rectangle  in  a  square  is  shown 
in  Fig.  32. 


A   H  & 


In  the  square  AB,  Fig.  32,  draw  the  line  CD,  dividing  the  square  into  two 
equal  parts.  Draw  ED,  the  diagonal  to  two  squares.  On  DG  describe  a  semi- 
circle. The  arc  of  this  cuts  the  line  ED  at  F.  Through  the  point  F  draw  the 
line  HI  parallel  to  GB.  The  area  HB  is  a  root-five  rectangle  within  the  area 
of  the  square  AB. 


DYNAMIC  SYMMETRY  41 

In  the  1 8th  proposition  of  the  thirteenth  book  a  diagram  is  furnished  which 
illustrates  the  setting  out  of  the  "five  figures"  for  the  purpose  of  comparison. 
The  "five  figures,"  of  course,  mean  the  five  regular  solids.  These  solids  were  of 
much  interest  to  the  Greeks  of  the  Sixth  Century  B.  C,  because  it  was  then 
thought  that  the  a^eifTs~6T~the  elements,  which  made  up  the  universe,  were 
shaped  like  the  tetrahedron,  the  octahedron,  the  cube,  and  the  icosahedron. 
The  dodecahedron  was  regarded  as  the  shape  which  encompassed  all  the 
others. 

The  basis  of  the  diagram  in  the  1 8th  proposition  of  the  13th  book  is  a  semi- 
circle on  a  given  line.  In  brief  the  operation  is  this: 


Fig.  33- 


AB  is  the  given  line  and  ABE  is  the  semicircle.  (See  Fig.  33.)  Euclid  in  sub- 
stance says:  at  A  draw  a  line  equal  to  AB  at  right  angles  to  that  line 
and  call  its  point  of  termination  G.  The  point  C  is  midway  between  A  and  B. 
Connect  C  and  G.  In  Euclid's  diagram  the  point  H  is  the  intersection  of  the 
line  GC  with  the  arc  of  the  semicircle  AEB.  From  H  a  line  is  drawn  parallel 


Fig-  34- 


to  AG  to  meet  AB  at  K,  BL  is  made  equal  to  AK.  From  the  point  L  a  line  is 
drawn,  parallel  to  AG  to  meet  the  arc  of  the  semicircle  at  M.  It  is  obvious  that 
HLKM  is  a  square  and  that  HA  and  MB  are  rectangles  of  the  whirling  squares. 
In  other  words,  Euclid  has  here  constructed  a  root-five  rectangle  and  defined 
the  square  in  the  center,  as  is  often  necessary  in  the  analysis  of  Greek  design. 


42  DYNAMIC  SYMMETRY 

Euclid  further  shows  in  this  proposition  that  the  comprehension  of  the  icosahe- 
dron  in  the  same  sphere  with  the  other  four  regular  solids  involves  the  side  of 
the  hexagon,  the  side  of  the  decagon  and  the  side  of  a  pentagon  inscribed 
in  the  same  circle.  AK,  BL  are  two  sides  of  the  decagon  and  KL,  KH,  LM  or 
HM  the  side  of  the  hexagon,  and  MB  is  the  side  of  a  pentagon. 

The  geometrical  constructions  used  by  Euclid  for  the  comprehension  of  the 
five  regular  solids  in  the  same  sphere,  suggest  another  method  of  determining 
the  root  rectangles  of  dynamic  symmetry  in  a  square.  This  method  is  based 
upon  the  fact  that  an  angle  in  a  semicircle  is  necessarily  a  right  angle. 


Fig.  35.  Fig.  36. 


The  simplest  example  of  this  is  shown  in  Fig.  35,  where  ABC  is  a  right  angled 
triangle.  B  is  also  the  center  of  the  square  AD. 

In  Fig.  36  the  line  CB  is  revolved  until  it  coincides  with  the  side  of  the  square, 
to  determine  the  point  E.  The  area  AEFC  is  a  root-two  rectangle.  It  will 
be  noticed  that  the  diagonal  of  the  reciprocal  of  the  root-two  rectangle  AEFC 
cuts  the  diagonal  of  the  whole  at  G,  and  that  this  point  lies  on  the  arc  of  the 
semicircle.  If  the  line  GC  is  revolved  until  it  coincides  with  CE  it  will  deter- 
mine the  point  for  a  root-three  rectangle.  The  poles  or  eyes  of  all  the  root 
rectangles,  that  is,  the  points  where  the  diagonals  of  their  reciprocals  cut  the 
diagonals  of  the  whole  will  lie  on  the  arc  of  this  semicircle  and  in  each  case  the 
lines  similar  to  GC  of  the  root-two  rectangle  will  determine  the  points  on  CE 
for  each  successive  rectangle.    Fig.  37  suggests  the  construction  for  this. 


Fig-  37- 


DYNAMIC  SYMMETRY  43 

The  geometrical  fact  established  by  Euclid  that  if  a  circle  is  described  with 
a  side  of  a  whirling  square  rectangle  as  radius,  this  line  equals  the  side  of  a 
hexagon,  the  end  of  the  rectangle,  the  side  of  a  decagon  and  the  diagonal  of 
the  rectangle,  the  side  of  a  pentagon,  all  inscribed  in  this  same  circle,  suggests 
the  construction  of  Figs.  39  and  40. 


Fig.  39.  Fig.  40. 


In  Fig.  39,  AC  is  the  diagonal  of  a  whirling  square  rectangle,  BC  the  end  and 
AB  the  side.  AD  is  the  side  of  a  pentagon  and  AE  is  the  side  of  a  decagon.  The 
line  DG  is  a  diagonal  of  a  pentagon  inscribed  in  the  circle,  and  it  cuts  the  side 
of  the  whirling  square  rectangle  at  H.  The  area  BH  is  equal  to  two  squares  and 
AH  is  composed  of  two  root-five  rectangles,  while  HM  is  equal  to  four  such 
shapes.  The  line  PI  passes  through  the  point  E  of  the  decagon.  AI  is  equal  to 
two  whirling  square  rectangles,  while  PC  is  equal  to  a  1.309  shape.  NL  is  an 
area  represented  by  the  ratio  2.1 18  or  1.618  plus  .5.  This  area  is  also  equal  to 
two  root-five  rectangles  plus  a  square,  1.118  plus  1.  JK  is  a  square  escribing 
the  circle  with  radius  AB  and  ML  is  a  whirling  square  rectangle  in  the  center. 
The  areas  MJ  and  LK  are  each  composed  of  two  whirling  square  rectangles 
plus  two  squares.  In  Fig.  40  AB  is  the  side  of  a  hexagon  equal  to  AC,  the  radius 
of  the  circle.  BD,  EF  are  sides  of  two  equilateral  triangles.  These  two  lines 
divide  each  of  the  four  whirling  square  rectangles  AH,  AG,  CI  and  CJ  into  two 
equal  parts.  The  area  DF  is  a  root-three  rectangle. 


CHAPTER  FOUR:  ROOT  RECTANGLES 
AND  SOME  VASE  FORMS 

NALYSES  of  Greek  and  Egyptian  compositions  show  that  the  artist 
always  worked  within  predetermined  areas.  The  enclosing 
rectangle  was  considered  the  factor  which  controlled  and  de- 
termined the  units  of  the  form.  A  work  of  art  thus  correlated 
became  an  entity,  comparable  to  an  organism  in  nature.  It 
possessed  an  individual  character,  instinct  with  the  life  of  design. 

Only  such  rectangles,  simple  or  compound,  were  used,  whose  areas  and  sub- 
multiple  parts  were  clearly  understood.  If  the  design  for  a  vase  shape  were  being 
planned  the  artist  would  consider  the  full  height  of  the  vessel  as  the  end  or  side 
of  a  certain  rectangle,  while  the  full  width  would  be  the  other  end  or  side.  The 
choice  of  a  rectangle  depended  upon  its  suitability  for  a  purpose,  both  in  shape 
and  property  of  proportional  subdivision.  A  rough  sketch  was  probably  made 
as  a  preliminary  and  this  formalized  by  the  rectangle.  Most  Greek  pottery 
shapes,  however,  were  traditional,  being  slowly  developed  through  a  long  period 
of  time;  consequently,  rough  sketches  of  ideas  must  have  been  rare.  From  gener- 
ation to  generation,  from  father  to  son,  craft  ideas  were  passed  along,  acquiring 
refinement  gradually. 

Modern  art,  as  a  rule,  aims  at  freshness  of  idea  and  individuality  in  tech 
nique  of  handling;  Greek  art  aimed  at  the  perfection  of  proportion  and  work- 
manship in  the  treatment  of  old,  well-understood  and  established  motifs.  That 
this  is  true  is  not  only  proven  by  the  standardized  shapes  of  Amphora,  Kylix, 
Kalpis,  Hydria,  Skyphos,  Oinochoe  and  Lekythos,  but  by  the  accepted  forms 
of  temples,  theaters,  units  of  decoration,  treatment  of  drapery,  grouping  of 
sculpture  forms  and  even  the  proportions  of  the  figure.  The  opportunity  for 
individual  expression  existed  only  in  superlative  workmanship,  in  refinement, 
precision  and  subtlety.  To  win  distinction  as  an  artist  it  was  necessary  for  the 
Greek  to  be  a  veritable  master.  The  danger  of  overrefinement  is  feared  by 
the  modern  artist,  for  it  has  become  a  tradition  that  this  leads  to  sweetness 
and  loss  of  virility,  because  it  invariably  ends  in  overwork  of  surfaces.  But  this 
peril  was  almost  unknown  to  the  ancient  Greek,  his  care  and  energy  were 
devoted  largely  to  the  refinement  of  the  structure  of  his  creations. 

Analysis  of  any  fine  Greek  design  is  sure  to  disclose  an  arrangement  of  area 
which  produces  the  quality  of  inevitableness,  so  conspicuously  absent  in  mod- 
ern art.  An  example  of  such  a  theme  is  furnished  by  a  handsome  red-figured 
amphora  of  the  Nolan  type,  in  the  Fogg  Museum  in  Boston.  Its  greatest  width 
divided  into  its  height  produces  the  ratio  of  1.7071.  This  ratio  shows  that,  as 


DYNAMIC  SYMMETRY 


45 


L 


i  / 
1  / 
y 

l>i" 

\l 

J 


Fig.  i.    Nolan  Amphora  in  the  Fogg  Museum  at  Harvard. 


an  area,  it  is  composed  of  a  square  plus  the  reciprocal  of  a  root-two  rectangle, 
i.  e.,  i.  plus  .7071,  the  fraction  being  the  square  root  of  two  divided  by  two. 

The  amphora  is  contained  within  the  area  of  a  root-two  rectangle  plus  a 
square  on  its  side.  The  width  of  the  lip,  in  relation  to  the  overall  form,  shows 
that  it  is  a  side  of  a  square  comprehended  in  the  center  of  the  root-two  rectan- 
gle. When  this  square  is  drawn  and  its  sides  produced  through  the  major 
square,  an  interesting  situation  exists  in  area  manipulation.  The  projection  of 
the  sides  of  this  square  through  the  major  square  produces  in  the  center  of  that 
square  a  root-two  rectangle  so  that  the  shape  as  defined  by  the  lip  is  a  square 
plus  a  root-two  rectangle,  Fig.  3^,  but  the  square  is  on  the  end  of  the  rectangle 
instead  of  on  the  side  as  it  is  in  the  major  shape.  The  method  of  simple  con- 


46 


DYNAMIC  SYMMETRY 


struction  by  which  the  figures  so  far  described  were  created  is  the  drawing  of  a 
square  and  its  diagonals.  (Fig.  3$.) 


Fig.  1. 

The  shaded  area  shows  the  rectangle  of  the  Amphora  design. 

The  side  of  the  root-two  rectangle  is  equal  to  half  the  diagonal  of  the  square. 
The  method  of  construction  by  which  the  secondary  square  and  root  two  are 
placed  within  the  major  shape,  is  shown  in  Fig.  4,  a,  b  and  c. 


\ 

\ 

\ 

\  y 

1  ! 

y 

1  1 

!  ! 

Fig.  '3a. 


Fig.  3b. 


A  root-two  rectangle,  AB,  is  cut  off  within  the  major  shape,  its  side  being 
made  equal  to  the  diagonal  of  the  major  square.  This  applied  rectangle  is  in 


Fig.  4«. 


Fig.  \b. 


Fig.  4f . 


DYNAMIC  SYMMETRY 


47 


"defect"  and  the  area  left  over  is  composed  of  two  squares  and  one  root-two 
rectangle,  as  shown  in  b,  Fig.  4.  The  same  construction  is  used,  working  from 
the  other  end  of  the  major  shape,  as  shown  in  Fig.  4^. 


Fig.  5. 

Through  the  centers  of  the  small  squares  on  each  corner,  lines  are  drawn  paral- 
lel to  the  sides  of  the  major  figure.  These  lines  determine  the  secondary  square 
and  root-two  rectangle,  shown  in  Fig.  5.  A  diagonal  to  this  secondary  shape 
determines  the  angle  pitch  of  the  lip,  and  its  thickness,  also  the  width  of  its 
base,  and  the  width  of  the  neck.  (See  Fig.  1.)  LK  is  this  line. 

The  foot  of  the  amphora  is  proportioned  by  the  small  root-two  figure  and  two 
squares  at  the  base.  DE  is  the  root-two  rectangle.  A  square  is  placed  in  the 
center  of  this  shape,  being  CB.  The  width  of  the  ring  above  the  foot  is  the  side 
of  this  square.  The  width  of  the  top  of  the  foot  exhibits  an  interesting  manipu- 
lation of  the  square  and  root-two  figures  at  the  base  of  the  design.  The  line  AB 
in  Fig.  1  brings  out  the  point.  AB  is  a  derived  root-two  rectangle,  and  its  diag- 
onal is  cut  at  J  by  a  line  through  the  point  I.  The  thickness  of  the  foot  and  its 
width  at  the  bottom  are  determined  by  the  diagonal  and  perpendicular  of  the 
root-two  shape  DE.  (Fig.  1.) 

The  thickness  of  the  ring  above  the  foot  is  established  by  the  line  AB,  in  Fig. 
6,  a  diagonal  to  a  square  and  a  root-two  rectangle,  intersecting  the  side  of  the 
square  at  C. 

1     1  n  zptv  "i 


Fig.  6. 


48  DYNAMIC  SYMMETRY 

Two  white  pyxides,  ladies'  toilet  boxes,  one  in  the  Museum  of  Fine  Arts, 
Boston,  and  one  in  the  Metropolitan  Museum,  New  York,  furnish  examples  of 
Greek  design  for  comparative  study.  These  two  examples  of  the  ancient  pot- 
ter's craft  are  exactly  of  the  same  overall  shape;  the  ratio  in  each  case  being 
i. 207 1.  This  is  a  compound  shape  composed  of  the  reciprocals  of  root  four  or 
half  a  square  and  root  two,  .5  plus  .7071.  The  reciprocal  of  1.2071  is  .8284, 
and  this  divided  by  two  equals  .4142,  or  the  difference  between  unity  and  the 
square  root  of  two,  1.4I42,  i.  e.,  the  square  root  of  two  minus  1.  When  a  square 
is  subtracted  from  a  root-two  rectangle  the  excess  area  is  composed  of  a  square 
and  a  root-two  rectangle. 


y~z 

S 

S 

*  .414-2     *  /.  x 

Fig.  7.  'ig.  8. 

The  containing  rectangle  of  each  pyxis  design,  therefore,  is  composed  of  two 
.4142  figures,  i.  e.,  two  squares  plus  a  root-two  rectangle.  (Figs.  7  and  8.) 

The  details  of  the  two  designs,  however,  are  proportioned  or  themed  differ- 
ently. In  the  Boston  example  the  line  AB  of  the  analysis  passes  through  the 
center  of  the  root-two  shape.  (Fig.  10.)  The  line  AB  is  the  top  of  the  pyxis. 

The  width  of  the  bowl  at  its  narrowest  point  is  equal  to  the  end  of  the  major 
root-two  rectangle,  i.  e.,  it  is  the  side  of  the  square  CD  constructed  in  the  cen- 
ter of  this  rectangle.  (Fig.  9.)  HI  is  a  diagonal  to  a  .4142  rectangle,  i.  e.,  half 
the  composing  shape.  This  line  cuts  the  diagonal  of  the  square  CD  at  J.  There- 
fore the  rectangle  JK  is  a  similar  shape  to  the  whole,  two  squares  and  a  root- 
two  rectangle,  and  is  the  containing  rectangle  of  the  knob.  LK  is  composed  of  a 
square  and  a  root-two  rectangle.  The  line  MN  is  a  side  of  the  square  MNOP. 

When  unity  is  applied  to  a  1.2071  rectangle  the  excess  area  is  composed  of 
two  squares  and  two  root-two  rectangles.  This  is  the  elevation  area  of  the  foot. 


A  WHITE-GROUND  PYXIS,  MUSEUM  OF  FINE  ARTS,  BOSTON 

{Compare  with  White-Ground  Pyxis  from  New  York) 
A  theme  in  root-two 


R  is  the  center  of  the  two  squares  of  the  base.  S  is  the  center  of  the  square  MP. 
A  further  refinement  in  the  design  is  shown  by  the  sinking  of  the  handle  below 
the  outer  rim  of  .the  cover.  The  only  variation  from  extraordinary  exactitude 
is  at  the  juncture  of  the  lid  shown  by  the  line  EF.  This  is  worn  at  the  edges  so 
that  it  is  difficult  to  determine  this  line  precisely.  The  error,  however,  is  so  small 
that  it  cannot  be  shown  in  the  drawing. 

This  pyxis  was  measured  and  drawn  by  Dr.  L.  D.  Caskey,  of  the  Boston 
"Museum  of  Fine  Arts. 

The  analysis  of  this  vase  shows  a  consistent  Greek  theme  in  area  and  it  may 
readily  be  seen  that  not  only  the  content  of  the  design  itself  but  the  excess  area 
not  occupied  by  the  design,  may  be  expressed  in  terms  of  the  whole  and  the  two 
composing  shapes,  namely,  the  root-four  and  root-two  reciprocals.  HO  is  a 


5o  DYNAMIC  SYMMETRY 

square,  HL  two  squares  and  a  root-two  rectangle.  The  application  of  this  area 
to  the  square  HQ  leaves  the  area  CL,  a  root-two  rectangle.  HA  is  a  root-two 
rectangle.  The  application  of  the  square  HO  leaves  the  area  CA,  a  square  and  a 
root-two  rectangle. 


Fig.  10.  Fig.  1 1, 


The  design  plan  of  the  pyxis  in  the  Metropolitan  Museum,  New  York,  de- 
pends upon  a  manipulation  of  the  diagonal  to  the  overall  shape  and  to  the 
two  composing  figures,  the  root-four  and  root-two  reciprocals.  The  manner 
in  which  this  is  done  discloses  an  interesting  feature  of  Greek  design  practice. 
It  seems  to  have  been  recognized  early  that  diagonals  were  the  most  important 
lines  in  the  determination  of  both  direct  and  indirect  proportions.  In  the  present 
example  diagonals  of  the  whole  intersect  diagonals  of  the  root-two  rectangle 
at  A  and  B,  Fig.  10.  Through  these  points  are  drawn  the  lines  HF,  EG,  IJ 
and  LK,  through  the  points  C  and  D.  These  lines  subdivide  the  area  of  the 
root-two  rectangle  into  squares  and  root-two  shapes.  CE,  AG  are  squares,  MC, 
DN,  AP  and  BO  are  root-two  rectangles.  AI  and  BJ  are  two  root-four  rectangles, 
i.  e.,  shapes  of  two  squares  each.  IJ  is  the  top  of  the  pyxis,  DH  the  square  en- 
closing the  handle  or  knob. 

AB  in  Fig.  1 1,  is  a  square,  one  side  of  which  is  the  width  of  the  bowl  at  the 
narrowest  point.  The  sides  of  this  square  produced,  determine  the  root-two 
rectangle  BC  and  fix  the  line  of  the  base  by  their  intersection  with  the  diagonals 
of  the  whole  at  the  points  D  and  E.  I 

The  intersection  of  the  diagonals  of  the  whole  with  the  diagonals  of  half  the 
major  shape,  at  AB  in  Fig.  12,  determine  the  thickness  of  the  lid. 


A  WHITE-GROUND  PYXIS,  METROPOLITAN  MUSEUM,  NEW  YORK 

{Compare  with  the  Boston  White-Ground  Pyxis) 
A  theme  in  root-two 


The  Fifth  Century  B.  C.  bronze  oinochoe,  Fig.  13,  99.485  in  the  Museum  of 
Fine  Arts,  Boston,  in  its  plan  scheme,  is  another  admirable  illustration  of  the 
Greek  method  of  arranging  a  theme  in  area.  The  jug  was  measured  and  drawn  by 
Dr.  Caskey,  before  an  analysis  of  the  shape  was  made.  The  containing  rectangle 
is  a  root-two  shape,  and  all  details  are  determined  by  a  consistent  arrange- 
ment of  the  elements  of  this  figure.  The  diagonals  and  perpendiculars  are  drawn 
to  the  overall  shape  and  a  square  described  in  the  center  of  the  root-two  figure 
AB.  This  square  is  CD,  the  side  of  which  is  equal  to  the  width  of  the  lip  of  the 
vase.  The  diagonals  of  the  whole  cut  the  sides  of  this  square  at  E  and  F.  This 
determines  the  area  CF,  equal  to  two  squares,  EG,  FH,  and  the  root-two  figure 
HI.  A  line  drawn  from  J  to  C  cuts  the  side  of  the  square  GE  at  K.  The  line 
KLM  divides  the  area  of  this  square  into  two  squares,  CL,  LI,  and  two  root- 
two  figures,  GL  and  LE.  The  center  of  the  square  CL,  fixes  the  top  of  the  lip; 


52 


Fig.  13.    Bronze  Oinochoe  in  the  Boston  Museum. 
(Measured  and  drawn  by  L.  D.  Caskey.) 

the  base  of  this  square,  ML,  establishes  the  bottom  of  the  lip.  Diagonals  and 
perpendiculars  to  the  root-two  figure  HI,  determine  other  proportions  of  the 
lip  and  handle  juncture.  Aline  drawn  through  the  center  of  the  root-two  figure 
BO,  establishes  the  two  root-two  figures  PO,  PQ.  The  width  of  the  vase,  at  the 
base,  is  fixed  by  the  centers  of  the  two  squares  SO,  RO.  The  sides  of  these 
squares  produced,  as  from  T  to  I,  cut  the  diagonals  of  the  whole  and  perpen- 
diculars, as  at  T  and  U.  This  fixes  the  figure  UV,  of  which  TW  is  a  square. 
Diagonals  to  half  the  area  of  this  square,  as  WX,  determine  the  triangle  in 
which  the  goats'  heads  are  drawn.  The  beard  of  one  of  these  heads  is  shorter 
than  that  of  the  other,  probably  due  to  the  molten  bronze  not  entirely  displac- 
ing the  wax  in  the  casting.  If  a  square  is  applied  to  the  other  end  of  the  shape 
occupied  by  the  heads  of  the  goats,  other  details  are  obtained.  This  design 
may  now  be  understood  as  a  theme  in  root-two  and  square.  The  drawing  was 
made  exactly  the  size  of  the  original  and  no  other  analysis  is  possible. 


DYNAMIC  SYMMETRY  53 

A  black-figured  kylix,  98.920  in  the  Boston  Museum  (Fig.  14),  fills  an  area 
composed  of  three  root-two  rectangles,  and  the  width  of  the  foot  is  the  end  of 
one  of  these  shapes.  AB  is  a  root-two  rectangle,  BC  is  a  square  applied  to  it,  CE 
is  a  diagonal  to  the  excess  area  or  to  a  square  plus  a  root-two  rectangle.  AF 
is  a  root-two  rectangle  and  its  diagonal  intersects  CE  at  D,  and  fixes  the  width 
of  the  bowl.  The  depth  of  the  bowl  is  determined  by  the  point  G,  the  intersec- 
tion of  a  diagonal  of  the  square  BC  with  the  diagonal  of  the  root-two  rectangle 
AB.  (Compare  with  Yale  Skyphos,  p.  62.) 


Fig.  14. 

(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


The  ratio  of  a  black-figured  kylix  from  Yale,  Fig.  15,  is  that  of  a  square 
plus  a  root-two  figure  or  1.4I42  plus  1.  In  this  case  the  square  is  drawn 
in  the  center  and  a  reciprocal  root-two  figure  on  either  end.  AB  is  the  side  of 
the  square.  C  and  D  are  the  intersections  of  diagonals  of  squares  and  root-two 
rectangles.  I  and  J  are  the  intersections  of  diagonals  to  two  figures,  each  com- 
posed of  a  root-two  rectangle  plus  the  large  square,  with  a  line  drawn  through 
the  middle  of  the  large  square,  and  G  and  H  are  the  intersections  of  these 
same  diagonals  with  the  diagonals  of  the  major  square.  The  consistency  of  the 
proportions  of  the  foot  in  relation  to  the  width  of  the  bowl  is  now  apparent. 
The  point  K  is  the  intersection  of  the  diagonal  of  the  whole  with  the  diag- 
onal of  a  square. 

An  Attic  black-figured  hydria,  95.62  in  the  Boston  Museum  (Fig.  16),  is  a  vase 
form  of  unusual  distinction.  The  plan  is  a  theme  in  root-two.  The  vessel  is  a  splen- 


54 


DYNAMIC  SYMMETRY 


Fig.  15.    Black-figured  Kylix  in  the  Stoddard  Collection  at  Yale. 


did  example  of  Greek  craftsmanship.  II  the  width  of  the  bowl  is  taken  as  the 
end  and  the  total  height  as  a  side  of  a  rectangle  the  ratio  is  1 .2071,  the  reciprocal 
being  .8284.  This  is  the  same  rectangle  as  that  of  the  pyxides  in  this  chapter. 
The  overall  ratio  obtained  by  including  the  handles,  is  1.0356.  This  rectangle 
is  simply  .8284  plus  .2071,  a  rather  ingenious  manipulation  of  shapes.  If  the 
fraction  .2071  be  divided  by  two  .10356  is  the  result.  This  means  that  the  area 
of  the  overall  rectangle  AB  is  the  1.2071  shape  which  is  composed  of  the  two 
squares  CD  and  DJ  and  the  root-two  rectangle  is  AJ.  The  lines  IJ  and  IC 
are  diagonals  to  the  reciprocals  of  AJ.  These  diagonals  intersect  the  diagonals 
of  the  1 .207 1  form  as  at  H.  The  line  OM  is  a  side  of  the  root-two  rectangle  MN. 
The  line  ST  bisects  the  areas  of  the  two  squares  CD,  DJ,  and  the  root-two 
diagonals,  as  MN,  cut  this  bisecting  line  of  the  two  squares  at  S  and  T.  This 
fixes  the  proportions  of  the  foot.  The  width  of  the  lip  is  the  side  of  a  square, 
PO,  in  the  center  of  the  root-two  rectangle  AJ.  The  handle  extends  above  the 
lip  and  the  root-two  rectangle  XY,  with  its  included  square  XZ,  shows  the  pro- 
portional relationship.  The  diagonal  GF  cuts  the  side  of  the  square  PO  at  A'. 
The  area  FA'  is  a  1 .2071  shape  and  H'  is  its  center.  FF'  equals  two  squares  and 
G'  is  the  center.  The  square  A'B'  is  described  on  the  side  of  A'F;  C  is  its 
center.  B'D'  is  a  root-two  rectangle  with  a  square  applied  to  the  end  to  es- 
tablish the  point  E'.  The  base  of  the  pictorial  composition  is  the  line  CJ,  the 
top  of  the  two  squares  CD,  DJ.  The  painted  rays  at  the  foot  terminate  at  the 
line  L'M'.  This  line  fixes  the  side  of  a  square  applied  to  AB,  i.  e.,  the  line  L'M' 
is  distant  from  the  top  of  the  containing  shape  an  amount  equal  to  GB.  The 
point  K',  which  marks  the  line  separating  the  two  pictorial  compositions,  is 
obtained  by  diagonal  to  the  shapes  PP'  and  O'N. 


DYNAMIC  SYMMETRY  55 


.     .10333         «  *   8a8«» 

«  ,   (.0353 


Fig.  16.    Boston  Black-figured  Hydria  95.62. 
(Measured  and  drawn  by  L.  D.  Caskey.) 

If  the  width  of  the  foot  is  considered  as  the  end,  and  the  full  height,  AG,  as 
the  side  of  a  rectangle,  it  will  be  a  2.2071  shape,  i.  e.,  two  squares  plus  .2071. 
The  area  value  of  this  fraction  is  two  squares  plus  two  root-two  rectangles. 
That  the  designer  of  this  vase  must  have  known  something  of  this  value  is 
evidenced  by  the  fact  that  the  rectangle  J'U  is  a  .2071  shape  and  the  height 
of  the  vase,  minus  the  foot,  is  equal  to  twice  the  width  of  the  foot. 

If  the  width  of  the  lip  is  considered  as  the  end  and  the  full  height,  AG,  as  the 
side  of  a  rectangle  the  ratio  for  the  shape  is  1.7071,  the  scheme  of  the  Fogg 
amphora  of  this  chapter. 

An  early  black-figured  kylix  in  the  Fogg  Museum,  at  Harvard,  has  the  same 
ratio  as  the  kylix  from  Yale  (see  Fig.  15),  i.  e.,  2.4142,  a  square  and  a  root-two 
figure.  The  method  of  subdivision  however  is  quite  different.  The  square  AB 
is  applied  to  the  root-two  figure  AC  and  its  base  line  produced  to  D.  This 
determines  the  root-two  figure  DE  in  the  square  EF.  The  excess  area  FB  is 


56 


DYNAMIC  SYMMETRY 


Fig.  17.    Black-figured  Kylix  from  the  Fogg  Museum,  Harvard. 


composed  of  two  squares  and  a  root-two  rectangle,  the  sides  of  which,  added, 
equal  the  width  of  the  foot.  The  square  CJ  in  the  root-two  rectangle  AC  de- 
termines the  area  LA,  a  square  and  a  root-two  rectangle.  The  square  EM 
fixes  the  area  NM,  also  a  square  and  a  root-two  rectangle.  The  diagonal 
NM  is  the  angle-pitch  of  the  lip  and  is  a  similar  angle  to  the  diagonal  of  the 


Fig.  18.    A  root-two  Oinochoe  from  the  Boston  Museum. 


A  BLACK-FIGURED  HYDRIA,  MUSEUM  OF  FINE  ARTS,  BOSTON 
A  theme  in  root-two.    There  is  no  break  in  the  sequence  of  the  theme 


DYNAMIC  SYMMETRY  57 

entire  figure.  The  area  KD  is  composed  of  two  squares.  BO,  OD  are  diagonals 
to  squares  and  root-two  rectangles.  OPOQ  is  a  root-two  rectangle.  RS  and 
RT  determine  the  angle  pitch  of  the  foot. 

A  red-figured  oinochoe  in  the  Boston  Museum,  Fig.  18,  is  a  simple  root-two 
rectangle.  A  and  B  are  poles  or  eyes  of  the  two  root-two  figures  MK  and  NL.  U 
and  V  are  eyes  to  the  major  or  overall  shape.  C  and  D  are  eyes  to  the  two  root- 
two  rectangles  GQ  and  HR.  GF  is  a  square,  JK  is  a  square.  The  decorative  band 
at  the  base  of  the  figure  composition  passes  through  the  center  of  the  square  RS 
while  a  side  of  the  square  GF  passes  through  the  compositional  band  at  the 
top  of  the  figures. 

A  Nolan  amphora  in  the  Stoddard  Collection  at  Yale,  Fig.  19,  duplicates  the 
ratio  1. 707 1  of  the  amphora  of  the  Fogg  Museum  at  Harvard.  The  division  of 


Fig.  19.    Nolan  Amphora  from  Yale. 


58  DYNAMIC  SYMMETRY 

the  area  however  is  somewhat  different.  AB  is  the  major  square  and  AC  the 
root-two  rectangle.  CD  is  a  square  in  the  root-two  rectangle  and  DE  is  the 
excess  area  equal  to  a  root-two  shape  and  a  square.  EF  is  this  square  and  EG 
is  a  root-two  rectangle  within  it.  The  center  of  the  root-two  area  HG  is  the 
point  which  fixes  the  proportions  at  the  juncture  of  lip  and  neck.  EI  is  a  similar 
shape  to  the  whole.  AX  is  a  diagonal  to  a  square  and  it  cuts  the  diagonal  of 
the  whole  at  J.  EM  is  a  root-two  rectangle  and  the  area  MN  is  composed  of 
two  squares  and  a  root-two  rectangle.  The  side  of  this  root-two  form  is  the 
width  of  the  foot  at  its  top.  OP  is  a  diagonal  of  a  shape  similar  to  the  whole, 
i.  e.,  a  square,  RN  plus  a  root-two  figure,  OR.  The  point  S  is  the  eye  of  the  area 
OR.  The  relation  of  the  point  T  to  the  foot  is  apparent.  The  angle  pitch  of 
the  foot  is  fixed  by  the  lines  KV  and  KW.  The  point  L  is  the  center  of  the 
major  square  and  a  factor  in  the  proportions  of  the  meander  band  under  the 
picture. 


r 


CHAPTER  FIVE:  PLATO'S  MOST 
BEAUTIFUL  SHAPE 

"""N^HE  Nolan  type  amphora,  here  illustrated,  13.188  in  the  Mu- 
seum of  Fine  Arts,  Boston,  is  an  example  of  a  vase  design  cor- 
related by  a  root-three  rectangle.  It  is  remarkable  that  this 
shape  is  not  more  often  met  with  in  Greek  design,  for  we 
know  that  it  was  regarded  as  a  beautiful  shape.  It  is  mentioned 


by  Plato,  who  makes  the  Pythagorean  Timaeus  explain:  "  'Each  straight  lined 
figure  consists  of  triangles,  but  all  triangles  can  be  dissected  into  rectangular 
ones,  which  are  either  isosceles  or  s'calene.  Among  the  latter  the  most  beautiful 
is  that  out  of  the  doubling  of  which  an  equilateral  arises,  or  in  which  the  square 
of  the  greater  perpendicular  is  three  times  that  of  the  smaller,  or  in  which  the 
smaller  perpendicular  is  half  the  hypotenuse  (in  length).  But  two  or  four 
right-angled  isosceles  triangles,  properly  put  together,  form  the  square;  two 
or  six  of  the  most  beautiful*  scalene  right-angled  triangles  form  the  equi- 
lateral triangle;  and  out  of  these  two  figures  arise  the  solids  which  correspond 
with  the  four  elements  of  the  real  world,  the  tetrahedron,  octahedron,  icosahe- 
dron  and  the  cube.'  "  (Quoted  by  Allman,  "History  of  Greek  Geometry  from 
Thales  to  Euclid,"  p.  38.)  Classic  art  was  practically  over  by  Plato's  time. 

The  relation  of  the  square  on  the  end  to  a  square  on  the  side  of  a  root-three 
figure  is  as  one  to  three,  while  the  end  is  one-half  the  length  of  the  diagonal. 
The  Greek  artists  do  not  seem  to  have  agreed  with  Plato  concerning  the  beauty 
of  this  rectangle,  for  we  find  it  but  seldom.  It  appears  occasionally  in  vases; 
and  the  double  equilateral  triangle  or  hexagon  appears  in  important  Greek  archi- 
tecture only  in  the  Choragic  Monument  of  Lysicrates.  The  equilateral  triangle 
is  one  of  the  two  fundamentals  of  static  symmetry  and  as  a  correlating  form  was 
used  lavishly  in  Saracenic  and  Gothic  art.  (See  chapter  on  Static  Symmetry.) 

Certainly  a  root-three  rectangle  cannot  be  said  to  be  more  beautiful  than 
any  of  the  other  shapes  of  dynamic  symmetry.  In  fact,  there  is  little  ground  for 
the  assumption  that  any  shape,  per  se,  is  more  beautiful  than  any  other. 
Beauty,  perhaps,  may  be  a  matter  of  functional  coordination. 

In  the  analysis  of  the  amphora  13.188  in  the  Boston  Museum,  Fig.  1,  per- 
pendiculars to  its  diagonals  indicate  the  divisions  of  a  root-three  rectangle 
into  three  similar  shapes  to  the  whole.  AB  is  a  root-three  rectangle  and  a 
reciprocal  of  the  major  shape,  as  are  also  AC,  CD,  EF,  and  G  is  the  center  of 
the  rectangle  CD. 

H  is  the  center  of  the  rectangle  AI.  JK  is  a  root-three  rectangle  and  L  and 
*  The  "most  beautiful"  oblong,  here  referred  to,  is  the  root-three  rectangle. 


6o 


DYNAMIC  SYMMETRY 


Fig.  i.    Nolan  Amphora  13.188  in  the  Boston  Museum. 
(A  theme  in  root-three.) 

M  are  its  eyes.  The  width  of  the  lip  is  fixed  by  the  points  O  and  P,  intersections 
of  the  sides  of  the  two  squares,  NK,  with  the  diagonals  of  the  root-three  rec- 
tangle JK.  A  very  slight  error  exists  at  O,  the  juncture  of  the  neck  and  bowl. 

Nolan  Amphora  01.8109  in  the  Boston  Museum,  Fig.  2,  picture  by  "the  Pan 
Master,"  is  a  root-three  rectangle.  AB  is  a  root-three  rectangle,  as  are  also  AC  and 
CD.  The  point  E  is  the  eye  of  the  root-three  rectangle  AB.  The  point  F  is  the 
center  of  the  root-three  rectangle  AE,  and  P  is  the  center  of  the  root-three 
shape  CD.  In  the  root-three  rectangle  at  the  base  of  the  overall  shape  the  point 
K  is  the  eye.  A  line  through  this  point  parallel  to  the  base  line  determines  the 
four  root-three  rectangles  IJ.  The  area  HM  is  a  root-three  rectangle,  as  is  also 


DYNAMIC  SYMMETRY 


61 


\   p  j 

E 

/ 

/  1 

..c  \1 

Fig.  2.    Nolan  Amphora  01.8109  in  the  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 

HL.  The  point  O  is  the  intersection  of  the  diagonal  of  a  square  on  the  base 
line  with  the  side  of  a  root-three  rectangle.  N  is  the  center  of  the  root-three 
rectangle  at  the  base  and  fixes  the  base  of  the  meander  band. 

A  small  cup  in  the  Stoddard  Collection  at  Yale  is  a  simple  root-three  rec- 
tangle divided  dynamically,  but  use  was  made  of  the  equilateral  triangle  in 
the  arrangement  of  the  three  feet.  These  feet,  however,  follow  the  diagonal 
of  the  secondary  root-three  forms.  The  width  of  the  base  is  the  end  of  a  root- 
three  rectangle  and  the  proportions  of  the  painted  bands  near  the  top  of  the 
bowl  are  clearly  shown  in  the  diagram.  (Fig.  3.) 

Skyphos  160  in  the  Stoddard  Collection  at  Yale,  Fig.  4,  is  a  root-three  shape 
and  the  detail  is  correlated  by  the  application  of  squares  on  either  end  of  the 


62 


DYNAMIC  SYMMETRY 


>'3L  /X 

Fig.  3.    Small  Cup  in  the  Stoddard  Collection  at  Yale. 
(A  theme  in  root-three.) 

rectangle.  The  width  of  the  bowl  is  determined  by  the  intersection  of  the  side 
of  the  square  AB  with  a  diagonal  of  the  square  CD,  see  point  H.  The  width  of 
the  foot  is  fixed  by  the  intersection  of  diagonals  to  the  squares  AB  and  CD,  as 
at  I.  A  line  from  I  to  C  intersects  a  side  of  the  square  CD  at  G  to  place  the  com- 
positional line  under  the  picture.  The  height  of  the  foot  is  the  intersection  of  a 
diagonal  to  half  the  entire  shape  as  FC  intersecting  the  diagonal  of  a  square. 


Fig.  4.    Skyphos  160  in  the  Stoddard  Collection  at  Yale. 


An  early  black-figured  hydria,  108  of  the  Stoddard  Collection  at  Yale,  Fig.  5, 
is  a  theme  in  root-three  and  squares.  The  overall  plan  is  composed  of  two  root- 
three  rectangles,  one  on  top  of  the  other,  AB  and  BC.  Squares,  as  CD,  AD,  BO 
and  FE,  are  applied  to  the  two  root-three  shapes  from  either  end.  They  overlap 
in  the  center  to  the  extent  of  FD.  The  overlapping  of  these  squares  has  the 


DYNAMIC  SYMMETRY 


63 


effect  of  dividing  the  entire  area  into  a  rectangular  pattern  or  mesh  propor- 
tioned by  root-three  rectangles.  This  is  a  remarkable  pattern  form  and  it  is 
strange  that  no  attempt  was  made  to  use  the  equilateral  triangles  which  are 
inherent  in  the  root-three  shapes.  The  center,  L,  of  the  square  AD  fixes  the 
width  of  the  foot.  The  side  of  the  square  AG  cuts  the  diagonal  of  the  square 
AD  at  N.  This  establishes  the  width  of  the  bowl  and  also  the  height  of  the  foot, 
as  is  apparent  at  M.  The  area  of  the  foot  elevation  is  composed  of  two  squares 
and  two  root-three  rectangles,  and  the  width  of  the  lip  at  its  base  is  fixed  by  a 
line  drawn  from  Q,  the  center  of  the  base  of  the  foot,  to  the  point  P.  It  seems  to 
have  been  intended  that  the  angle  pitch  of  the  lip  should  fall  outside  the  point 
P  because  the  full  width  of  the  lip  at  its  base  is  equal  to  one-half  the  height  of 
the  vase,  that  is,  it  is  the  side  of  a  square  placed  in  the  center  of  the  root-three 
rectangle  BC.  The  point  K  is  the  center  of  the  square  IJ.  This  point  has  two 
functions;  it  establishes  the  line  which  separates  the  two  pictures  and  is  im- 
portant in  fixing  the  lip  proportions. 

If  the  width  of  the  foot  is  considered  as  an  end  and  the  full  height  as  the  side 


64  DYNAMIC  SYMMETRY 

of  a  rectangle,  the  ratio  is  2.732,  i.  e.,  a  root-three  rectangle,  1.7321,  plus  1. 
The  area  made  by  the  width  of  the  bowl  and  the  full  height  has  the  ratio  1.366. 
The  fraction  .366  is  equal  to  .732  divided  by  two.  The  point  U,  through  which 
passes  the  juncture  of  neck  and  bowl,  is  the  center  of  the  rectangle  ST.  The 
lip  thickness  is  fixed  by  a  line  from  C  to  U  and  the  width  of  the  neck  at  its 
juncture  with  the  bowl  by  a  line  from  C  to  S. 


CHAPTER  SIX:  A  BRYGOS  KANTHAROS 
AND  OTHER  POTTERY  EXAMPLES  OF 
SIMILAR  RECTANGLE  SHAPES 

<A  STRIKINGLY  beautiful  kantharos  of  the  Fifth  Century  B.  C,  now 
/\\        in  the  Museum  of  Fine  Arts  at  Boston,  furnishes  an  admirable 
/    \  \       example  of  the  use  of  a  compound  shape  derived  from  a  root- 
J        \\     five  rectangle.  The  area  of  the  enclosing  shape  has  an  end  to 
A.  )  V    side  relationship  of  i :  i.i  1 8.  The  ratio  i.i  1 8  multiplied  by  two 

equals  2.236,  the  square  root  of  five.  The  ratio  may  be  stated  as  root-five  divided 
by  two.  A  root-five  rectangle  divided  by  two,  or  cut  in  half,  is  composed  of  two 
root-five  rectangles  one  over  or  one  beside  the  other.  The  heavy  lines  of  the 
diagram  define  this  shape.  The  area  AB  is  the  overall  rectangle  of  the  kantharos. 


TS 

2 

c 

A 

Fig.  1. 


This  area  is  that  part  of  a  square  denned  by  the  pentagon  and  shown  in 
Fig.  6a,  Chapter  III. 

The  double  root-five  shape  may  be  subdivided  in  many  ways  to  produce 
themes  in  abstract  form.  The  primary  subdivision  would  be  that  of  each  com- 
posing rectangle  into  a  square  and  two  whirling  square  rectangles.  And,  be- 
cause this  is  a  vase  elevation  design  with  elements  symmetrically  disposed, 
the  two  squares  would  be  constructed  in  the  center  as  in  Fig.  2.  The  entire  area 


h/  s 

S 

ws 

S 

Fig.  2. 


would  be  divided  into  two  squares  and  four  whirling  square  rectangles.  The 
designer  of  the  kantharos,  however,  used  but  one  element  of  this  arrangement. 


66  DYNAMIC  SYMMETRY 

This  element  is  the  side  of  one  of  the  squares,  which  is  employed  to  establish 
the  strongly  emphasized  line  AB.  (See  Fig.  3.) 


Fig.  3. 

The  arrangement  of  the  area  which  constitutes  the  selected  theme,  depends 
upon  the  application  of  four  whirling  square  rectangles  constructed  upon  the 
four  sides  of  the  rectangle.  These  applied  rectangles  overlap  and  produce  the 
pattern  shown  in  Fig.  4. 


 1   k  t 

Fig.  4. 

The  whirling  square  rectangles  are  AB,  CD,  DE  and  BF.  The  areas  AC,  FE, 
are  the  important  features  of  the  design.  The  area  AC  determines  the  width 
of  the  bowl  and  FE  the  width  of  the  stem  at  its  juncture  with  the  bowl. 


Fig-  5- 


DYNAMIC  SYMMETRY 


67 


In  Fig.  5,  AD  is  the  containing  rectangle  of  the  kantharos.  The  lines  AB  and 
BC,  at  the  points  of  their  intersection  with  the  side  of  a  whirling  square  rec- 
tangle, fix  the  width  of  the  bowl.  These  lines  are  the  diagonals  to  the  two  halves 
of  the  major  shape,  consequently  the  elevation  of  the  kantharos,  either  with  or 
without  its  handles,  is  a  double  root-five  rectangle. 


IV  s 


Fig.  6. 

AD  is  a  double  root-five  rectangle,  as  is  also  FE.  And  either  of  these  shapes 
will  furnish  an  analysis  of  the  design.  When  a  whirling  square  rectangle  is  ap- 
plied to  the  side  of  a  double  root-five  rectangle,  the  area  on  the  line  AB,  as  in 
Fig.  6,  is  composed  of  two  squares  and  a  whirling  square  rectangle.  The  recipro- 
cal of  1. 118  is  .8944;  this  is  the  line  CB.  CA  equals  .618  and  AB,  .2764.  AB  is 
the  difference  between  .618  and  .8944  or  .2764;  this  fraction  .2764  divided  into 
unity  equals  3.618.  AB  is  the  reciprocal  of  two  squares  plus  a  whirling  square 
rectangle,  2  plus  1.6 18.  Within  the  major  rectangle,  therefore,  the  excess  area, 
at  both  the  top  and  the  bottom  of  the  bowl,  is  composed  of  two  squares  and  a 
whirling  square  rectangle  (see  Fig.  7),  and  AB  is  a  whirling  square  rectangle 
and  C  is  its  eye. 


Fig.  7. 


This  point  C  fixes  the  width  of  the  foot.  The  analysis  is  now  complete,  or 
is  carried  as  far  as  is  necessary. 

Dr.  Caskey  shows  in  his  drawing  of  the  kantharos,  Fig.  8,  the  exact  error 
in  the  handle  adjustment.  The  adjustment  of  these  delicate  handles  must  have 
been  a  problem  because,  even  if  the  vase  left  the  potter's  hand  perfectly  fixed, 


68 


DYNAMIC  SYMMETRY 


Fig.  8.    Kantharos  in  the  Boston  Museum. 
(Measured  and  drawn  by  L.  D.  Caskey.) 


he  could  never  tell  how  much  shrinkage  in  baking  would  disorganize  his  plans. 
In  this  case,  however,  the  error  of  the  handles  makes  no  difference  because  the 
bowl  is  a  similar  shape  to  the  whole.  The  writer  has  found  that  the  small  errors 
found  in  Greek  pottery,  except  in  few  cases,  are  practically  negligible.  This  is 
true  for  the  reason  that  a  part  of  a  design  which  has  been  dynamically  pro- 
portioned is  always  some  recognizable  submultiple  of  some  recognizable  rec- 
tangle. Therefore  it  is  really  better  to  make  the  small  corrections  necessary  to 
true  up  an  example.  In  his  drawing  of  this  kantharos,  the  actual  discrepancy 
appears  at  the  top  of  the  drawing.  The  width  of  the  handles  is  correct,  and 
when  the  double  root-five  rectangle  is  drawn,  its  side  is  the  mean  between  the 
handle  heights.  The  writer's  drawing  of  this  vase  is  shown  in  Fig.  9  with  the 
handle  discrepancy  corrected. 

When  two  whirling  square  rectangles  are  applied  to  the  sides  of  a  double 
root-five  rectangle,  as  in  the  case  of  this  Brygos  kantharos,  they  overlap.  The 
area  of  the  "overlap"  is  determined  thus.  If  the  side  of  this  shape  is  used  as 
unity  then  the  end  is  .8944,  the  reciprocal  of  1.118.  The  reciprocal  of  a  whirling 
square  rectangle  is  .618.  This,  subtracted  from  .8944,  leaves  .2764  which,  again, 
is  the  reciprocal  of  3.618  and  is  the  area  on  either  side  of  the  "overlap."  The 
reciprocal  .2764  subtracted  from  .618  equals  .3416.  This  represents  the  overlap 


IN  THE  WRITER'S  OPINION  THIS  KANTHAROS  IS  ONE 
OF  THE  FINEST  OF  GREEK  CUPS 

A  theme  in  double  root-five 


DYNAMIC  SYMMETRY 


69 


A- /'       '•  \ 

M  

\\\ 

///'  /  

\  1 

1 

T^^  ;  : 

s  ; 

Fig.  9.    Drawing  of  the  Boston  Kantharos  with  Handles  Corrected. 


and  is  the  reciprocal  of  2.927.  This  ratio  is  compound  and  consists  of  two  rec- 
tangles, as  1.1 18  plus  1.809.  It  is  now  clear  that  this  overlap  area  consists  of  a 
double  root-five  rectangle  and  a  square  plus  two  whirling  square  rectangles. 
The  ratio  1.809  is  one  of  the  basic  shapes  of  the  pentagonal  form  and  consists 
of  a  square  plus  a  whirling  square  rectangle  divided  by  two. 

Again,  this  "overlap"  area  may  be  considered  as  1.6 18  plus  1.309,  i.  e.,  a 
whirling  square  rectangle  plus  a  square  and  two  reciprocals  of  such  a  shape, 
.618  divided  by  two  equalling  .309. 


S  S  CV  3  S 


Fig.  10.  Fig.  11. 


The  area  of  the  elevation  of  KalpisG.  R.  591,  Fig.  12,  Metropolitan  Museum, 
New  York,  is  composed  of  two  root-five  rectangles,  of  which  AB  is  one.  The  width 
of  the  lip  is  fixed  by  the  point  F,  the  intersection  of  a  side  of  the  square  BC 
with  the  diagonal  ED  of  the  root-five  shape.  By  construction  the  area  FD  is  a 
root-five  rectangle,  while  AF  is  composed  of  two  1.382  shapes  or  the  ratio  2.764  in 


7o  DYNAMIC  SYMMETRY 


Fig.  12.    Kalpis  G.  R.  591,  Metropolitan  Museum,  New  York. 
(A  double  root-five  theme.) 

the  two  whirling  square  rectangles  AH  and  CH.  The  point  G,  fixing  the  width  of 
the  foot,  is  the  intersection  of  a  diagonal  to  the  whirling  square  rectangle  HE 
with  the  diagonal  of  the  root-five  rectangle.  The  angle  pitch  of  the  lip  is  a  line 
drawn  from  I  to  B.  The  angle  pitch  of  the  foot  is  shown  at  K,  which  is  found  by 
a  line  drawn  to  one  corner  of  the  square  BC.  It  will  be  noticed  in  this  example 
that  the  lines  showing  the  subdivisions  of  the  foot  and  lip  are  projected  until 
they  meet  diagonals  to  certain  shapes  drawn  from  the  corners  A  and  E.  This 
procedure  is  one  which  enables  the  eye  to  grasp  quickly  the  proportional  re- 
lationship which  exists  in  the  composing  units  of  a  Greek  design.  The  projec- 
tion of  the  first  subdivision  of  the  lip  intersects  the  diagonal  of  a  whirling  square 
rectangle  drawn  from  the  corner  A.  From  this  intersection  the  line  turns  at 
right  angles  and  is  carried  downward  until  it  intersects  the  diagonal  of  a  square 
drawn  from  the  corner  E.  Here  it  meets  the  projection  of  the  first  division  of 
the  foot.  This  tells  us  that  the  first  division  of  the  lip  is  related  to  the  first  divi- 
sion of  the  foot  on  the  proportion  of  a  whirling  square  rectangle  to  a  square,  a 
fact  which  is  not  immediately  obvious  by  construction.  Again,  the  base  of  the 


DYNAMIC  SYMMETRY 


71 


lip  is  projected  until  it  meets  the  diagonal  of  a  square  drawn  from  A.  From  this 
point,  at  right  angles,  it  is  carried  until  it  intersects  the  diagonal  of  a  square 
drawn  from  the  corner  E,  where  it  meets  the  projection  of  the  top  of  the  foot. 
We  may  see  by  this  that  the  lip  and  foot  are  the  same  thickness  because  their 
projections  both  meet  the  diagonal  of  a  square.  Also  it  is  apparent  that  the 
width  of  the  bowl  is  related  to  both  foot  and  neck  on  the  proportion  of  a  square. 

The  two  bands  at  the  base  of  the  pictorial  composition  are  determined  by 
the  points  L  and  M.  The  diagonal  of  the  square  BC  meets  the  diagonal  of  the 
whole  at  L.  The  diagonal  of  the  square  BC  meets  the  diagonal  of  two  squares 
at  M. 

This  kalpis  shows,  unmistakably,  that  the  picture  is  secondary.  The  shape 
of  the  vessel  is  determined  with  great  care  while  the  picture  is  ordinary.  Even 
the  height  of  the  male  figure  is  miscalculated,  as  he  is  not  standing  on  the  same 
level  with  the  female  figure.  The  hands  of  the  female  figure  and  the  right  arm 
and  hand  of  the  male  figure  are  badly  drawn. 

Two  root-five  rectangles  furnish  the  overall  shape  for  Kalpis  08.417  in  the 
Boston  Museum,  Fig.  13.  The  width  of  the  bowl  as  an  end  and  the  height  of  the 


Fig.  13.    Kalpis,  08.417,  Boston  Museum. 


72 


DYNAMIC  SYMMETRY 


vase  as  a  side  supplies  a  1.309  rectangle,  i.  e.,  a  square  and  two  whirling  square 
rectangles.  AB,  CE  are  squares  and  AD,  BL  are  double  whirling  square  rec- 
tangles. When  the  .309  shape  as  AD  is  applied  to  the  square  AB  the  excess 
area  BE  is  a  square  plus  a  root-five  rectangle  or  the  ratio  .691.  EF  is  the  square 
and  FD  is  the  root-five  rectangle.  The  relation  of  the  double  root-five  shape  to 
the  1.309  rectangle  is  shown  by  the  lines  CH  and  GK,  which  are  diagonals  to 
the  whole.  The  points  H  and  J  show  this  connection.  The  point  J  is  connected 
with  an  important  element  of  the  foot  of  the  vase. 

When  a  whirling  square  rectangle  is  applied  to  the  end  of  a  double  root-five 
rectangle  the  excess  area  consists  of  the  reciprocal  of  a  root-four  rectangle, 
i.  <?.,  .5  or  two  squares,  .618  plus  .5  equals  1. 118. 

Whirling  square  rectangles  applied  to  both  ends  of  a  double  root-five  rec- 
tangle, overlap.  The  area  of  this  "overlap"  is  the  difference  between  .5  and 
.618  or  .118  plus.  This  fraction  will  be  recognizable  as  an  area  if  we  consider 
it  as  .236  divided  by  two.  .236  is  the  difference  between  root-four  and  root-five, 


DYNAMIC  SYMMETRY  73 

i.  e.,  2.236  minus  2.  It  is  the  reciprocal  of  4.236  or  two  whirling  square  rec- 
tangles plus  a  square,  1.618  multiplied  by  two  plus  one.  .118  is  the  reciprocal 
of  two  such  shapes  lying  end  to  end  or  8.472  or  four  whirling  square  rectangles 
plus  two  squares. 

It  must  be  stressed  repeatedly  that  these  curious  areas  which  are  found  so 
abundantly  in  nature  and  in  Greek  art,  cannot  be  too  carefully  studied.  And 
for  that  purpose  it  is  necessary  to  have  recourse  to  arithmetic.  We  must  re- 
member that  we  are  dealing  with  forms  of  design  used  by  the  best  artists  and 
craftsmen  the  world  has  known,  who  worked  without  stint  of  labor  for  gener- 
ations. If  we  followed  the  steps  of  the  Greeks  and  acquired  our  knowledge  of 
these  shapes  entirely  by  geometrical  construction,  the  labor  would  be  too  great 
for  an  ordinary  lifetime.  By  using  arithmetic  in  conjunction  with  geometrical 
construction,  an  ordinary  student  may  acquire  a  working  use  of  dynamic 
symmetry  in  a  few  months. 

The  double  root-five  rectangle  is  found  in  two  kalpides  in  the  Boston  Mu- 
seum, Nos.  91.224  and  91.225.  The  plan  scheme  of  the  first,  91.224,  Fig.  14,  is 
simple.  AB  is  a  whirling  square  rectangle  applied  to  the  end  of  the  shape  and 
CD,  CE  are  two  squares.  AJ  and  HI  are  two  whirling  square  rectangles  which 
overlap  to  the  extent  of  the  width  of  the  foot.  The  lip  width  is  clear.  FG  is  a  line 


Fig.  15.    Drawing  of  large  Volute  Krater  in  the  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


74  DYNAMIC  SYMMETRY 

through  the  center  of  the  overall  shape.  The  points  K  and  L  lie  on  the  diagonals 
to  the  two  squares  CD  and  CE.  The  points  M  and  N  are  intersections  of  the 
diagonals  of  the  whole  with  the  diagonals  of  the  whirling  square  rectangle. 

The  arrangement  and  proportioning  of  detail  in  kalpis  91.225  differs  but 
little  from  that  of  91.224  except  in  size,  the  latter  being  slightly  larger. 

A  large  volute  krater  in  the  Boston  Museum,  90.153,  is  composed  in  a  double 
root-five  rectangle.  AB,  BC  are  the  two  root-five  shapes  and  GE  is  a  square.  A 
slight  error  is  shown  at  the  top  where  the  handle  volutes  exceed  the  containing 
area.  The  large  square,  however,  and  the  complete  theme  in  the  arrangement  of 
detail,  justify  the  analysis.  The  width  of  the  foot  as  end  and  the  height  of  the 
bowl  furnish  another  root-five  rectangle  of  which  SU  is  the  square  and  OS,  UR 
are  two  whirling  square  rectangles. 


LARGE  BRONZE  HYDRIA,  METROPOLITAN  MUSEUM,  NEW  YORK 
One  of  the  most  carefully  worked  designs  in  existence 


V 


I 


Fig.  i.    Bronze  Hydria,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


CHAPTER  SEVEN:  A  HYDRIA, 
A  STAMNOS,  A  PYXIS  AND 
OTHER  VASE  FORMS 

BRONZE  hydria,  06.1078  in  the  New  York  Museum,  Fig.  1,  supplies 
a  ratio  of  1.045.  ^e  rati°  2.045  appears  in  a  handsome  bronze 
oinochoe  in  the  Boston  Museum.  The  1.045  area  occurs  fre- 
quently in  Greek  pottery.  It  is  composed  of  a  whirling  square 
rectangle,  .618  plus  .427.  The  fraction  .427  appears  in  the  pen- 
tagon form  (see  Chapter  III  as  .854,  i.  e.,  .427  multiplied  by  two).  .809  plus  .236 
also  equals  1.045.  In  this  example  it  is  clear  that  the  designer  had  this  sub- 
division in  view  because  the  area  BD  is  this  ratio,  i.  <?.,  two  whirling  square  rec- 
tangles. BC  is  a  square  and  JM  is  a  root-five  rectangle  in  thecenterof  this  square; 
that  is,  the  vase  without  the  lip  is  a  square.  The  end  of  this  root-five  rectangle 


76  DYNAMIC  SYMMETRY 

is  the  width  of  lip  and  foot.  The  area  AG  is  .236.  This  fraction  is  the  reciprocal 
of  4.236,  i.  e.,  a  root-five  rectangle  plus  two  squares,  2.236  plus  two.  The  area 
EF  is  composedof  the  two  squares.  AE  and  FG  are  together  equal  to  a  root-five 
rectangle.  IP  is  a  whirling  square  rectangle  and  the  construction  for  the  width 
of  the  bowl  is  shown  at  the  point  O  in  the  whirling  square  rectangle  IN. 

This  vase  and  the  root-two  themed  oinochoe  of  Chapter  IV  are  the  only 
bronze  examples  of  Greek  design  used  in  this  book.  The  percentage  of  error  is 
much  smaller  in  the  bronzes  than  in  the  pottery.  In  this  example  the  error  is 
astonishingly  small. 


Fig.  1.    Large  Stamnos  10.2 10. 15  in  the  Metropolitan  Museum. 
(Measured  and  drawn  by  the  Museum  Staff.) 

A  beautiful  large  stamnos,  10.210. 15,  Metropolitan  Museum,  New  York,  Fig.  2, 
has  a  ratio  of  1.1826.  This  is  a  shape  which  is  not  uncommon.  It  is  a  compound 
form  of  two  elements,  each  of  which  is  .5913,  this  ratio  being  the  reciprocal  of 
1. 69 1.  AB  is  a  square,  the  side  of  which  is  equal  to  half  the  overall  shape.  BC 
is  a  square  and  CD  is  a  root-five  rectangle.  AE  is  a  root-five  rectangle.  The 
relation  of  the  details  of  the  foot  to  the  whirling  square  rectangle  in  AE  is  ap- 
parent. The  square  Cp  is  divided  into  the  root-five  GH  and  the  whirling  square 
rectangle  CI.  KD  is  a  whirling  square  rectangle  in  the  square  JD.  In  the  double 
whirling  square  rectangle  CL  the  sides  of  the  two  squares  ML  produced 
through  K  determine  the  whirling  square  rectangle  in  the  square  JD.  This  line 
fixes  the  width  of  the  bowl. 

Thearea  .382plus two  whirlingsquare rectangles  (.809),  that  is,  1.191,  is  found 


A  LARGE  STAMNOS,  METROPOLITAN  MUSEUM,  NEW  YORK 
A  vase  showing  unusual  design  power 


DYNAMIC  SYMMETRY  77 

in  a  pyxis  92.108  in  the  Boston  Museum.  AB,  or  the  vase  without  its  cover,  is  the 
.809  ratio.  AC,  or  the  area  of  the  cover,  is  the  .382  shape.  The  point  H  is  the  eye 
of  the  whirling  square  rectangle  AJ.  Consequently,  the  area  of  half  the  foot  is  a 
root-five  rectangle  and  the  whole  area  of  the  foot  is  equal  to  two  such  shapes 
or  a  root-twenty  rectangle.  F  is  the  eye  of  the  whirling  square  rectangle  KE. 
DE  is  a  whirling  square  rectangle.  AM  is  a  whirling  square  and  Nis  its  center. 
Dr.  Caskey's  small  drawing  shows  clearly  the  composing  units  of  the  area. 


Fig.  30.  Fig.  3^. 

R.  F.  Pyxis  in  the  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


Nolan  amphora  10.184,  Boston  Museum,  has  an  area  ratio  1.691,  Fig.  4. 
This  is  a  shape  which  appears  in  the  angle  column  adjustment  of  the  Parthenon. 
AB  and  AC  are  squares.  CD  is  a  root-five  rectangle.  CO  is  a  whirling  square 
rectangle  in  the  root-five  shape  CD.  EB  and  GF  are  two  root-five  rectangles. 
FI  is  a  1.809  shape  and  HI  is  its  major  square.  GK  is  a  2.618  shape.  The  line 
which  marks  the  juncture  of  neck  and  bowl  is  equal  in  length  to  the  width  of  the 
foot.  The  diagonal  to  the  square  AC  is  used  to  fix  the  length  of  this  line.  Its 
relation  to  the  root-five  shape  GF  is  shown  at  U. 

Nolan  amphora  136  in  the  Stoddard  Collection  at  Yale,  Fig.  5,  has  an  overall 


78  DYNAMIC  SYMMETRY 


Fig.  4.    Nolan  Amphora  10.184,  Boston  Museum. 

shape  of  1.809  and  the  bowl  width  is  exactly  one  half  the  height.  The  meander  at 
the  base  of  the  composition  is  placed  on  the  half  division  of  the  square  AB.  The 
foot  is  exactly  one-half  the  width  of  the  square  AB.  The  whirling  square  rec- 
tangle intersection  of  the  diagonal  of  AH  with  the  diagonal  of  LM  determines 
the  juncture  of  neck  and  body.  The  rectangle  NO,  which  encloses  the  foot,  is 
composed  of  two  root-five  shapes.  IJ  is  equal  to  the  width  of  the  foot. 

Nolan  amphora  01.18  in  the  Boston  Museum  apparently  has  the  same  ratio 
as  01.16,  Fig.  6,  a  and  b.  The  paintings  seem  to  be  by  the  same  hand.  In  this 
example,  however,  there  is  a  slight  error  as  shown  by  the  handles.  The  two  vases 
have  exactly  the  same  height  but  the  bowl  of  01.18  is  wider  than  01.16.  Other- 
wise the  proportions  are  nearly  the  same. 

The  ratio  1.809  appears  in  amphora  01.16.  This  area  is  composed  of  a  square 


DYNAMIC  SYMMETRY  79 

plus  two  whirling  square  rectangles.  FD  is  a  square  and  DE  two  whirling 
square  rectangles.  AB  is  a  whirling  square  rectangle,  as  is  also  HL.  The  subdivi- 
sion of  HL  and  its  relation  to  the  proportions  of  the  lip  and  neck  are  clear.  The 
point  C  is  on  the  diagonal  to  the  area  DE.  The  point  M  is  on  the  diagonal  to 
the  square  LB.  The  proportions  of  the  foot  in  relation  to  the  whirling  square 
area  PQ  are  also  clear.  The  width  of  the  bowl  in  this  example  is  just  half  the 
height  of  the  vase,  i.  e.,  minus  the  handles  the  area  of  the  vessel  is  exactly  two 
squares.  Compare  the  Nolan  amphora,  Metropolitan  Museum,  New  York, 
Fig.  8,  in  this  chapter. 


Fig.  5.    Nolan  Amphora  136  at  Yale. 


Nolan  Amphora  12.236.2,  Metropolitan  Museum,  New  York,  Fig.  7,  has  an 
overall  shape  of  1.764.  The  fraction  .764  is  the  reciprocal  of  1.309.  In  the  arrange- 
ment of  the  units  of  the  composition  AB  is  a  square,  DC  is  a  square,  and  BC  is 


8o  DYNAMIC  SYMMETRY 


Fig.  6a.  Fig.  6b. 

Two  Nolan  Amphorae  in  the  Boston  Museum,  01.16  and  01.18. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


.309  or  two  rectangles  of  the  whirling  squares.  EI  is  4.236,  or  two  rectangles 
of  the  whirling  squares  plus  a  square.  The  bowl  juncture  with  the  foot  is  a  side 
of  this  small  square.  The  area  GH  equals  the  area  BC.  The  proportions  of  the 
neck  and  lip  are  apparent.  The  width  of  the  bowl,  with  the  total  height  is  the 
ratio  1.927,  the  fraction  .927  being  .309  multiplied  by  3.  The  ratio  1.764  may 
also  be  considered  as  .882  multiplied  by  two,  and  also  as  the  ratio  2.764  minus 
one.  The  ratio  2.764  is  equal  to  a  square  plus  root-five  multiplied  by  four,  i.  e., 
.691  x  4. 

Nolan  Amphora  1 2.236.1,  Metropolitan  Museum,  New  York,  on  page  82,  Fig. 
8,  is  the  rectangle  1.854  (see  various  skyphoi).  This  ratio  is  obtained  by  mul- 
tiplying .618  by  3.  The  proportional  details  are  so  clear  that  explanation  is 


DYNAMIC  SYMMETRY 


81 


unnecessary.  The  point  A  is  the  intersection  of  a  diagonal  of  the  whole  with  the 
diagonal  of  a  whirling  square  rectangle.  The  line  DE  cuts  a  rectangle  of  the 
whirling  squares  from  the  square  BC. 

The  width  of  the  bowl  divides  the  total  height  into  two  squares,  i.  e.,  the 
width  of  the  bowl  is  half  the  height  of  the  vase.  The  point  F,  the  base  of  the 
meander  band  under  the  composition,  is  the  center  of  one  of  these  squares. 
The  error,  due  to  distortion,  is  shown  by  the  lip  at  the  top  of  the  rectangle. 

The  scheme  of  the  large  dinos  and  stand,  page  83,  is  a  square  plus  a  root-five 
rectangle,  the  ratio  being  1.4472,  the  reciprocal  .691.  This  is  a  monumental  piece 
of  pottery  and  the  theme  of  the  design  is  worth  careful  study.  The  general  shape 
appears  repeatedly  in  both  archaic  and  classic  Greek  art  and  is  the  basic  motif  in 


Fig.  7.    Nolan  Amphora  12.236.2  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


82  DYNAMIC  SYMMETRY 

the  plan  of  the  Parthenon.  The  general  theme  in  this  case  is  a  division  of  1.4472 
by  four.  One-fourth  of  1.4472  is  .3618.  This  ratio  of  .3618  is  the  reciprocal  of 
2.764.  One-fourth  of  2.76415.691,  which  is  the  reciprocal  of  1.4472,  i.e.,  a  square 
and  a  root-five  rectangle.  If  the  area:of  1.4472  is  divided  by  four,  both  side  and 
end,  sixteen  squares  and  sixteen  root-five  rectangles  result.  If  the  width  of  the  foot 
is  considered  as  the  end  of  the  rectangle  AB,  this  rectangle  is  composed  of  four 


Fig.  8.    Amphora  12. 236.1  Metropolitan  Museum,  New  York. 
(Drawn  and  measured  by  the  Museum  Staff.) 

root-five  rectangles,  AE,  EF,  FG  and  GH,  or  the  ratio  1.7888,  i.  e.,  .4472  multi- 
plied by  four,  and  each  root-five  rectangle  is  constructed  in  the  center  of  a  2.764 
rectangle.  If  the  width  of  the  lip  is  considered  as  the  end  of  the  rectangle  CD, 
this  shape  is  composed  of  four  root-four  rectangles.  A  root-four  rectangle  is 
composed  of  two  squares,  and  each  root-four  rectangle  is  constructed  in  the 
center  of  a  root-five  rectangle  and  a  2.764  rectangle.  If  the  bowl  of  the  dinos  is 


A  DINOS  AND  STAND,  MUSEUM  OF  FINE  ARTS,  BOSTON 
A  design  theme  in  square  and  root-five 


DYNAMIC  SYMMETRY 


83 


Fig.  9.    Large  Dinos  and  Stand  in  the  Boston  Museum. 
(Measured  and  drawn  by  L.  D.  Caskey.) 


considered  as  a  rectangle,  apart  from  the  pedestal,  this  shape  is  composed  of 
two  whirling  square  rectangles.  If  the  pedestal  is  considered  as  an  area  it  is 
the  rectangle  1.045,  a  fairly  common  shape  in  classic  art.  The  combinations  of 
proportions  in  this  vase  might  be  amplified  to  cover  the  entire  fabric  of  Greek 
design.  This  is  also  a  good  example  of  the  free  use  of  ornament  within  the 
severe  limits  of  a  general  shape.  The  decorated  bands  on  the  bowl  and  pedestal 
are  loosely  rendered. 

A  ratio  which  frequently  appears  directly  or  indirectly  in  Greek  vase  designs  is 
1.472,  as  in  this  example  of  an  amphora  from  the  Boston  Museum,  Fig.  11. 


84  DYNAMIC  SYMMETRY 


Fig.  10.    Development  of  the  plan  theme  of  the  Boston  Dinos. 

This  area  may  be  subdivided  in  various  ways  but  the  method  employed  by 
Dr.  Caskey  in  his  analysis  is,  in  all  probability,  the  right  one.  It  is  .618  plus 
.618  plus  .236.  AB  is  one  .618  shape,  CD  is  the  other,  while  the  area  AD  rep- 
resents .236.  The  width  of  the  lip  as  the  end  and  the  total  height  of  the  vase  as 
side  is  a  2.382  rectangle.  By  the  same  method,  using  the  width  of  the  foot  as 
an  end,  the  rectangle  is  2.944  or  1.472  multiplied  by  two.  The  width  of  the  lip 


Fig.  11.    Amphora  from  the  Boston  Museum. 
(Drawn,  measured  and  analyzed  by  L.  D.  Caskey.) 


A  BLACK-FIGURED  AMPHORA  FROM  THE  BOSTON  MUSEUM 

These  early  black-figured  vases  rank  among  the  best  designs  the  Greeks  ever 
made.    The  adjustment  of  the  human  motif  to  the  shape  theme  is  superb 


DYNAMIC  SYMMETRY 


85 


Fig.  12.    A  Perfume  Vase  in  the  Boston  Museum. 
(Measured  and  drawn  by  L.  D.  Caskey.) 

is  also  the  side  of  a  square  constructed  in  the  whirling  square  rectangle  CD. 
This  is  shown  by  the  point  F,  the  center  of  the  whirling  square  rectangle  DE. 
The  effect  upon  the  proportions  of  the  top  of  the  vase  of  the  diagonal  to  this 
square  QR  is  shown  by  the  points  S  and  T.  The  whirling  square  rectangle  LP, 
by  construction,  establishes  the  proportions  of  the  foot. 

The  shape  of  a  perfume  vase,  Fig.  12,  is  interesting  because  it  shows  that 
the  design  of  the  bowl  and  lid  was  planned  by  two  separate,  but  proportional, 
rectangles.  The  area  occupied  by  the  bowl  and  foot  is  shown  by  the  rectangle 
RO,  and  consists  of  the  square  NO,  plus  the  square  RM  and  the  rectangle 
MP,  which  is  composed  of  two  root-five  rectangles.  The  ratio  is  1.472.  The 
fraction  .472  equals  .236  x  2  and  is  the  reciprocal  of  2.1 18  or  root-five  divided 
by  two  plus  one.  The  shape  of  the  rectangle  of  the  lid  and  handle  is  4.236, 
the  reciprocal  of  which  is  .236  or  two  squares  plus  a  root-five  rectangle.  This 
shape  equals  half  of  the  area  NP.  A,  C,  B,  E,  D  are  squares.  NO  is  a  square 
and  PQ  is  a  square.  G,  H  and  I  are  areas  represented  by  root-five,  2.236  divided 
by  two.  JK  is  a  root-five  rectangle.  LM  is  a  whirling  square  rectangle  in  the 
square  RM.  Every  detail  of  the  vase  may  be  expressed  in  terms  of  the  major 
shape. 

An  early  black-figured  krater,  07.286.76,  Fig.  13,  in  the  Metropolitan  Museum, 
New  York,  is  an  illustration  of  extreme  distortion  in  a  classic  shape.  One  side  of 


86 


DYNAMIC  SYMMETRY 


the  lip  is  much  higher  than  the  other  and  this  irregularity  exists  also  in  the  neck 
and  top  of  the  bowl.  Otherwise  the  vase  is  normal.  This  distortion,  probably, 


Fig.  13. 


happened  in  baking.  The  widths  at  all  points  are  true  with  a  center  line.  If  the 
width  of  the  bowl  be  taken  as  the  side  of  a  square,  this  square  is  shown  in  the 
drawing  as  DE,  and  if  the  sides  of  the  square  be  produced  to  the  extremities 
of  the  handles,  as  A  and  C,  then  the  areas  AB,  BC,  become  two  whirling 
square  rectangles.  The  analysis  need  not  be  further  extended  as  here  exists 
evidence  that  the  design  is  dynamic  and  the  general  distortion  is  shown  at  G, 
where  one  side  of  the  lip  extends  outside  the  encompassing  rectangle.  The 
other  side  of  the  lip  is  correct  and  the  overall  shape  is  1.1382.  Without  the  lip 
it  is  1.236.  The  lip  therefore  represents  the  difference  between  these  two  ratios. 
The  decimal  fraction  .1382  appears  in  the  Parthenon,  where  the  overall  rec- 
tangle of  the  ground  plan  is  2.1382. 

The  area  of  Kalpis  90.156,  Fig.  14,  a,  b,  c,  Boston  Museum,  is  composed  of 
a  whirling  square  rectangle  and  a  root-five  rectangle.  Dr.  Caskey's  two  small 
diagrams,  14^  and  14c,  show  the  general  proportions. 

Greek  symmetry,  as  has  been  pointed  out,  is  connected  with  the  geometrical 
properties  of  the  five  regular  solids  (see  Chapter  V),  and  the  proportions  of 
these  solids  are  associated  with  the  phenomena  of  leaf  distribution  in  Nature, 
therefore  it  is  not  unreasonable  to  expect  to  find  in  examples  of  that  sym- 


DYNAMIC  SYMMETRY  87 

metry,  such  as  are  furnished  by  temple  plans,  decoration,  bronzes  and  pottery, 
areas  and  subdivisions  of  areas  which  echo  and  re-echo  the  shapes  derivable 
from  the  regular  solids  and  the  summation  series  of  phyllotaxis.  That  this  is 
so  the  pottery  designs  alone  abundantly  show.  The  area  of  the  elevation  of  a 
Greek  vase  of  the  first  class,  that  is,  the  area  obtained  by  the  full  height  and 
width  of  such  a  vessel,  and  the  secondary  areas  obtained  by  subdivision  of  de- 
tails, such  as  width  of  foot,  neck,  lip  and  bowl  and  the  height  of  such  members, 
produce  a  series  of  shapes  which  could  not  be  obtained  accidentally.  This  is 
clearly  disclosed  by  the  analysis  of  a  large  krater,  10. 185  in  the  Boston  Museum 


Fig.  14A 


Fig.  14c. 


88 


DYNAMIC  SYMMETRY 


Fig.  15.    Large  Bell  Krater  10.185  with  lug  handles  in  the  Boston  Museum. 
(Drawn,  measured  and  analyzed  by  L.  D.  Caskey.) 


(Fig.  15).  The  height  of  this  krater  divided  into  its  width,  produces  the  ratio 
.882.  The  ratio  .882  is  composed  of  two  squares  on  top  of  a  .382  rectangle. 
The  .382  ratio  may  have  its  composing  elements  arranged  in  various  ways. 
For  example,  a  square  may  be  placed  in  the  center  and  double  whirling 
square  rectangles  on  either  side  as  in  No.  3  of  the  group  of  small  diagrams 
of  the  vase  made  by  Dr.  Caskey.  The  plan  scheme  of  this  krater  shows  that 
its  maker  possessed  a  high  order  of  design  knowledge,  particularly  in  de- 
termining and  arranging  similar  figures.  The  lines  AB,  BC  are  diagonals 
to  half  the  overall  shape;  at  D  and  E  they  cut  the  sides  of  the  square  FG,  this 
square  being  obtained  in  the  analysis  by  the  width  of  the  bowl.  The  rectangle 
DG,  that  is,  the  vessel  without  its  lip,  is  a  similar  shape  to  the  whole.  At  H  and 
I  these  two  diagonals  cut  a  line  drawn  through  the  center  of  the  major  shape. 
The  area  HO  is  a  similar  shape  to  the  whole.  HI  is  also  the  width  of  the  foot. 
The  area  OJ  is  a  shape  similar  to  double  the  shape  of  the  whole,  and  the  width 
of  the  foot  is  one  half  the  width  of  the  whole,  that  is,  the  area  QJ  is  expressed 
by  the  ratio  1.764,  this  being  .882  multiplied  by  two.  The  square  FG  bears  a 
ratio  relationship  to  the  width  of  the  vase  of  1.1708,  the  reciprocal  of  this  being 
.854.  The  line  KQ,  divided  into  the  height,  also  produces  the  ratio  1.1708  and 
LQ  is  the  square  on  the  end  of  this  shape.  If  two  squares  are  defined  in  the 


A  LARGE  BELL  KRATER  WITH  LUG  HANDLES, 
MUSEUM  OF  FINE  ARTS,  BOSTON 

This  vase  is  known  as  the  Actaeon  Krater 


DYNAMIC  SYMMETRY 


89 


.882  shape  HQ,  the  base  of  the  meander  band  MN  is  fixed  and  the  area  NQ  is  a 
.382  rectangle.  The  points  0  and  P  are  intersections  of  the  diagonals  of  the 
square  FG  with  the  sides  of  the  rectangle  QJ.  This  rectangle  is  also  connected 


v  

s 

s 

w 

s 

yv 

w 

C  w 

Fig.  16.    Dr.  Caskey's  analysis  of  the  Bell  Krater,  showing  the  com- 
posing units  of  form. 

with  the  1.309  rectangle  because  the  fraction  of  the  ratio  1.764,  i.  e.,  .764,  is  the 
reciprocal  of  1.309.  If  a  square  on  the  end  be  applied  to  the  area  QJ  the  excess 
would  be  a  1.309  shape.  The  area  of  an  .882  rectangle,  as  subdivided  by  the 
details  of  a  well-known  and  high-class  Greek  vase,  is  now  sufficiently  clear  for 
the  artist  and  designer. 

For  the  benefit  of  the  reader  unskilled  in  the  technique  of  design  the  point 
is  stressed  that  the  basic  facts  pertaining  to  the  area  occupied  by  a  composition 
are  paramount.  The  average  layman,  when  analyzing  a  design,  almost  invari- 
ably looks  for  an  aesthetic  motive,  some  arrangement  of  elements  which  creates 
a  pattern  or  movement  for  example.  This  is  a  fallacy,  for  the  reason  that  such 
effects  are  always  due  to  personal  selection  or  disposition  and  consequently 
cannot  be  taught  except  superficially.  The  facts  connected  with  areas  and 
volumes,  however,  are  impersonal,  are  general,  may  be  exhaustively  analyzed 
and  successfully  taught. 

Red-figured  Krater  07.286.81,  Fig.  17,  Metropolitan  Museum,  New  York,  a 
large  vase,  furnishes  the  same  ratio  as  the  two  pyxides  described  in  Chapter  IV, 
i.  e.,  1. 2071,  thecomprisingfigures  being  two  squares  plus  a  root-two  rectangle,  .5 


90  DYNAMIC  SYMMETRY 

plus  .7071.  In  the  root-two  rectangle,  AB,  DC  are  squares;  AC,  DB,  are  rec- 
tangles, each  consisting  of  a  square  plus  a  root-two  rectangle.  The  two  small 
squares  and  their  subdivisions  which  fix  the  proportions  of  both  foot  and  neck, 
and  the  dotted  line  which  shows  the  relationship  of  the  foot  to  the  neck,  do 
not  need  explanation. 


Fig.  17.    A  Bell  Krater  07.286.81  in  the  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


_  A 


Fig.  r.    Black-figured  Amphora  in  the  Boston  Museum. 
(Measured  and  drawn  by  L.  D.  Caskey.) 


CHAPTER  EIGHT:  FURTHER 
ANALYSES  OF  VASE  FORMS 

HANDSOME  black-figured  amphora,  13.76  in  the  Boston  Mu- 
seum, Fig.  1,  has  a  ratio  of  1.528  (compare  Lekythos  G.  R.  589 
New  York  Museum,  page  137).  The  fraction  .528  equals  a  square 
plus  two  root-five  rectangles.  AC  is  the  .528  shape.  This  fraction 
is  the  reciprocal  of  1.8944.  The  square  is  DE.GDand  CFare  two 
root-five  shapes.  The  diagonals  to  a  square  and  a  root-five  shape  intersect  at 
L  and  M.  The  area  00  is  a  whirling  square  rectangle  and  KN,  PC  are  two 
squares.  The  centers  of  these  squares  fix  the  width  of  the  lip.  The  area  KC  is 
a  .382  shape  and  GK  is  one-fourth  of  CB.  AB  is  the  major  square.  HB  is  a 
similar  shape  to  KC.  The  areas  HI,  JB  are  two  whirling  square  rectangles  and 


92  DYNAMIC  SYMMETRY 


Fig.  2.    Pelike  03.793,  Boston  Museum. 


IJ  is  a  1.382  shape.  HR  is  a  1.309  shape  or  a  square  plus  two  whirling  square 
rectangles.  The  area  JD  has  the  ratio  .472. 

A  red-figured  pelike,  03.793  in  the  Boston  Museum,  Fig.  2,  furnishes  a  ratio 
of  1.309.  AB  is  a  major  square.  CD  and  HI  are  whirling  square  rectangles.  The 
points  of  construction  in  these  shapes  are  clear.  The  width  of  the  foot  in  rela- 
tion to  the  height  is  1 :  1.854.  .618  x  3  =  1.854.  The  relation  of  this  foot  to  the 
lip  is  shown  by  the  line  NO. 

Attic  red-figured  pelike,  G.  R.  580,  Metropolitan  Museum,  New  York,  Fig.  3. 

This  vase  furnishes  a  ratio  of  1.309,  a  square  plus  two  whirling  square  rec- 
tangles, .309  being  .618  divided  by  2.  The  design  is  unusually  simple  and  it 
supplies  an  excellent  example  for  detailed  inspection. 

The  1 .309  shape  is  subdivided  by  two  whirling  square  rectangles  overlapping, 
as  AE,  GD,  so  as  to  produce  an  area  in  the  center  of  the  major  form  the  side  of 
which  is  equal  to  the  width  of  the  lip  of  the  vase.  The  relation  of  the  width  of 
the  foot  to  the  lip  is  apparent.  This  area  in  the  center  of  the  1.309  shape  is  ex- 
pressed by  the  ratio  2.1 18,  a  fairly  common  form  in  Greek  design.  Arithmeti- 
cally, this  ratio  may  be  written  2.236,  or  root-five,  divided  by  2,  or  1.118;  to 
this  ratio  a  square  or  unity  is  added,  making  2.1 18.  This  area  may  also  be 
described  as  two  squares,  or  2  plus  .118.  In  the  analysis  of  the  vase  it  will  be 
observed  that  this  arrangement  of  two  squares  plus  a  small  fraction  was 


Fig.  3.    Red-figured  Pelike  G.  R.  580,  Metropolitan  Museum,  New  York. 

actually  used  because  the  fraction  .118  expresses  the  area  of  the  elevation  of 
the  foot;  the  area  FE  in  the  analysis  being  root-four  or  two  squares. 

Consider  the  rectangle  1.309,  divided  as  described,  and  without  reference  to 
the  vase.  Draw  a  rectangle  of  the  whirling  squares  as  in  Fig.  4. 

AB  is  the  reciprocal  of  the  shape.  AC  is  a  square  in  this  reciprocal  and  a 
diagonal  of  this  square  cuts  the  diagonal  of  the  whole  at  D,  this  being  the 
point  which  determines  the  overlap  of  the  whirling  square  rectangles,  as  in 
the  analysis.  See  Fig.  5. 


Fig.  4.  Fig.  5.  Fig.  6. 


94  DYNAMIC  SYMMETRY 

AB  is  a  whirling  square  rectangle,  as  is  also  CD,  and  AE  is  a  1.309  shape, 
Fig-  5- 

This  construction  furnishes  a  remarkable  arrangement  in  proportion  as  will 
be  seen  in  Fig.  7,  where  the  proportional  subdivisions  are  briefly  indicated. 

This  vase  also  furnishes  a  good  example  for  arithmetical  analysis.  In  Fig.  6, 
the  length  AB  equals  1.618,  BC  1,  DC  .382,  BD  .618. 

By  geometrical  construction  a  line  through  the  point  G  cuts  the  whirling 
square  rectangle  EF  from  the  square  BF;  consequently  there  are  two  rec- 
tangles of  the  whirling  squares,  side  by  side,  EF  and  FC.  But  the  length  DC 
equals  .382,  therefore  ED  equals  .382  and  .382  multiplied  by  two  equals  .764 
and  this  fraction  divided  into  1.6 18  equals  2.1 18,  the  area  of  the  shape  CE  in 
the  analysis  of  the  vase.  This  arithmetic  method  may  be  readily  applied  to  any 
construction  or  analysis,  provided  the  larger  units  are  known,  as,  of  course,  they 
aways  are  in  dynamic  symmetry. 


areas: 


Fig.  7. 

.  7  the 

1.309  is 

subdivided  into  the 

following  series  of  proportiona 

AB  = 

1.236 

CG  =  44 

CJ  =  1.236 

CD  = 

2. 118 

FK  =  3.236 

IJ  =  2.618 

ED  = 

4/4 

GH  =  WS 

AK  =  Sq. 

BF  = 

4  4 

Al    =  *-3°9 

HJ  =  I-3°9>  etc- 

DYNAMIC  SYMMETRY  95 


Fig.  8.    Pelike  06.1021. 191,  Metropolitan  Museum,  New  York. 

A  large  simple  pelike,  06.1021. 191  in  the  Metropolitan  Museum,  New  York, 
Fig.  8,  is  a  theme  in  the  often  occurring  rectangle  1.382.  This  vase  supplies 
material  which  sheds  considerable  light  on  Greek  design  practice. 

The  width  of  the  lip  considered  as  the  end  of  a  rectangle,  of  which  the  full 
height  of  the  vessel  is  the  side,  defines  the  area  of  a  root-five  shape.  The  end  of 
this  rectangle  is  also  the  width  of  the  bottom  of  the  foot  of  the  vase.  At  some 
stage  of  development  the  design  probably  looked  like  the  diagram  in  Fig.  9. 

AB  is  a  1.382  rectangle,  CD  is  a  root-five  rectangle  in  the  center  of  the  major 
shape.  The  short  curved  lines  inside  this  latter  rectangle  at  the  top  and  outside 
at  the  bottom,  suggest  respectively  the  lip  and  foot. 

The  direct  subdivision  of  a  1.382  rectangle  is  shown  in  Fig.  10  where  AB  and 
CD  are  the  two  squares  described  on  the  ends  of  the  shape.  AD  and  CB  are 
two  .382  shapes  and  AE  is  a  rectangle  of  the  whirling  squares. 

When  a  root-five  rectangle  is  applied  to  the  center  of  this  containing  shape, 
as  in  Fig.  11,  the  major  area  is  subdivided  in  an  interesting  manner.  AB,  CD 
are  two  whirling  square  rectangles,  AE,  BF,  CG  and  DH  are  each  composed 


96 

A  

r~ 


L 

Fig.  9.  Fig.  10.  Fig.  11. 

of  two  squares,  while  EI,  and  the  similar  shape  on  the  other  side  of  the 
square  BI  are  each  double  whirling  square  rectangles.  BI  is  a  square  in  the 
center  of  the  whirling  square  rectangle  AE,  Fig.  10.  Considered  arithmetically 
the  major  area,  as  affected  by  the  root-five  shape,  is  as  follows: 

The  reciprocal  of  1.382  is  .7236.  If  the  side  FH,  Fig.  11,  represents  unity, 
then  the  end  HJ  represents  .7236.  In  relation  to  this  fraction,  the  end  of  the 
root-five  rectangle  CK  is  expressed  by  .4472,  and  this  fraction  subtracted  from 
.7236  leaves  .2764.  Dividing  this  by  2  the  fraction  .1382  is  obtained.  Thus  the 
areas  AJ  and  KF  are  each  composed  of  ten  similar  shapes  to  the  whole,  or  ten 
1.382  rectangles.  The  ratio  of  the  ground  plan  of  the  Parthenon  is  2.1382,  /.  e., 
it  is  composed  of  two  squares  plus  a  rectangle  similar  to  AJ  or  KF  of  this  pelike 
design.  The  fraction  .1382  may  be  further  identified  as  the  difference  between 
.309  and  .4472  or  a  root-five  shape  minus  two  whirling  square  rectangles.  The 
diagram,  Fig.  12,  shows  this  relationship. 


E  "  F 

Fig.  12. 


DYNAMIC  SYMMETRY 


AB  is  a  root-five  rectangle  with  the  square  FG  in  the  center.  AF,  ED  are  two 
whirling  square  rectangles,  as  are  also  AE,  FD.  The  shape  CB  is  a  .1382  rec- 


DYNAMIC  SYMMETRY 


97 


tangle  and  represents  the  difference  between  the  root-five  rectangle  AB  and 
the  double  whirling  square  area  AH.  The  meander  bands,  which  define  the 
limits  of  the  pictorial  composition,  are  related  to  the  general  proportion  of  the 
1.382  rectangle. 


Fig.  13.  Fig.  14. 


When  a  1.382  rectangle  is  divided  into  two  parts,  as  in  Fig.  14,  each  half  is 
composed  of  a  square  plus  a  root-five  figure.  The  bottom  of  the  meander  band 
at  the  base  of  the  figure  composition  passes  through  the  center  of  this  square. 
The  .382  area  of  a  1.382  rectangle  is  composed  of  a  square  plus  a  whirling 
square  rectangle,  see  Fig.  13. 

AB  is  the  whirling  square  rectangle.  AC  is  its  major  square  and  D  is  the  inter- 
section of  diagonals  to  these  two  shapes.  This  point  marks  the  top  of  the  mean- 


K 

/ 

t — 

A  N 

\  ! 
M  \/ 

/ 

A 

1 

Fig.  15. 


98  DYNAMIC  SYMMETRY 

der  band  above  the  figure  composition.  Fig.  15  shows  the  geometrical  method 
for  constructing  a  root-five  shape  in  the  center  of  a  1.382  rectangle.  AB  is  a 
.382  figure  and  C  and  D  are  the  centers  of  the  two  squares.  EF  is  a  root-five 
rectangle.  r 

Black-figured  Amphora  06.1021.69  in  the  Metropolitan  Museum,  New  York, 
Fig.  16,  has  a  ratio,  with  the  handles,  of  1 .3455  and  without  the  handles,  1.382. 
The  fraction  .3455  is  one-fourth  of  1.382.  The  width  of  the  lip  is  the  end  of  a 
root-five  rectangle  of  which  the  height  of  the  vase  is  the  side.  The  end  of  a  root- 
five  rectangle,  of  which  the  side  is  1.382,  is  represented  numerically  by  .618. 
The  width  of  the  foot  is  the  end  of  a  2.472  rectangle  described  in  the  center 
of  the  1.382  shape.  This  rectangle  is  composed  of  four  whirling  square  rec- 
tangles; .618  multiplied  by  4  equals  2.472.  CG  is  one  of  these  .618  rectangles. 
The  compositional  band  at  the  base  of  the  panelled  picture,  GH,  is  midway 
between  the  top  and  bottom  of  the  vase.  The  line  EF  is  one-fourth  the  total 
height.  The  angle  pitch  of  the  lip  is  determined  by  a  line  drawn  to  the  center 
of  the  foot,  or  the  diagonal  of  a  root-twenty  rectangle. 


Fig.  t6.    Black-figured  Amphora  06.1021.69,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


DYNAMIC  SYMMETRY 


99 


Fig.  17.    Psykter  in  the  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


A  psykter  in  the  Boston  Museum,  Fig.  17,  has  a  1.2764 shape.  (Seekalpis  in  this 
chapter.)  The  fraction  .2764  is  the  reciprocal  of  3.618.  In  Dr.  Caskey's  analysis 
AB  and  CD  are  whirling  square  rectangles.  AE  is  also  one  and  CF  is  the  3.618 
and  AF  a  square. 


lOO 


DYNAMIC  SYMMETRY 


The  ratio  of  1.0225  appears  in  an  early  black-figured  kalpis,  06.1021.69  in  the 
New  York  Museum,  Fig.  18.  This  shape  is  composed  of  .618  plus  -4045,  the  latter 
fraction  being  the  reciprocal  of  2.472  or  .618  multiplied  by  four.  The  rectangle 
contains  a  whirling  square  rectangle  plus  four  such  shapes  standing  on  top  of  it. 
The  width  of  the  bowl,  however,  with  the  height  of  the  vase  is  a  1.309  rec- 
tangle, i.  e.,  a  square  AB  plus  two  whirling  square  rectangles,  AC.  It  will  be 
noticed  that  the  side  of  the  square  AB  coincides  with  the  neck  and  bowl  junc- 
ture. T,  ON  and  L  are  points  which  fix  compositional  divisions  in  the  painting. 
SR,  RC  are  two  whirling  square  rectangles.  The  diagonal  SR  cuts  GF  produced 
at  T.  The  diagonals  of  the  two  whirling  square  rectangles  AC  proportion  the 
lip  and  neck  at  F  and  G.  The  whirling  square  rectangle  IJ  fixes  foot  propor- 
tions at  K.  The  line  NO  relates  the  foot  to  the  painted  band  under  the  pic- 
ture. AD  is  a  square. 


1  -\ 

Fig.  19.    Kalpis  in  the  Metropolitan  Museum,  New  York. 
(Drawn  and  measured  by  the  Museum  Staff.) 


The  red-figured  kalpis,  06. 102 1 . 1 90,  Fig.  1 9,  Metropolitan  Museum,  New  York 
City, has  a  major  shape  of  an  exact  square.  The  width  of  the  bowl  divided  into  a 
side  of  the  major  form  produces,  exactly,  a  1.309  rectangle.  The  simple  geomet- 
rical constructions  incident  to  the  comprehension  of  a  1.309  figure  in  the  cen- 


A  RED-FIGURED  KALPIS  IN  THE  METROPOLITAN  MUSEUM, 

NEW  YORK 
A  handsome  design  within  a  square 


DYNAMIC  SYMMETRY 


101 


ter  of  a  square  and  the  resultant  combinations  of  form  are  shown  in  the  small 
diagrams.  It  is  significant  that  the  angle  pitch  of  the  lip  is  the  diagonal  of  a 
.309  rectangle,  i.  e.,  ML,  Fig.  19,  is  a  .309  rectangle.  The  width  of  the  lip  as 
shown  by  NL  is  one-half  the  width  of  the  bowl.  The  width  of  the  neck  at  its 
narrowest  point  is  equal  to  the  width  of  the  juncture  of  the  foot  with  the  body. 


Fig.  20. 

Fig.  1  shows  the  geometrical  method  of  constructing  a  1.309  shape  in  the 
center  of  a  square.  AB  is  a  whirling  square  rectangle  comprehended  in  a  square. 
The  diagonals  of  two  squares,  CD  and  DE,  cut  the  side  of  the  whirling  square 
shape  AB  at  F  and  G. 

Fig.  2.  EC  is  a  1.309  rectangle.  AB  is  the  diagonal  to  two  squares.  DF  is  a 
square  and  DE  two  whirling  square  rectangles.  The  point  G  fixes  the  two  com- 
posing elements  of  the  1.309  rectangle. 

Fig.  3.  A  1.309  rectangle  is  divided  into  two  parts.  Each  part  is  composed  of 
a  square  plus  a  square  and  two  root-five  shapes. 

Fig.  4.  A  whirling  square  rectangle  applied  to  a  1.309  rectangle  leaves  a 
square  plus  a  root-five  shape. 

Fig.  5.  The  construction  for  the  lip  angle  of  the  kalpis,  AB  and  DC  are  .309 
shapes.  The  remaining  area  in  the  center  of  the  square  is  a  .382  shape. 

Fig.  6  is  a  .691  shape  applied  to  a  square.  The  .309  remainder  is  divided  into 
two  shapes,  one  being  .191  and  the  other  .118. 

Analysis  of  design  for  symmetry  is  slow  and  often  difficult.  Especially  is  this 


102 


DYNAMIC  SYMMETRY 


true  of  Greek  designs.  The  first  step  is  the  approximate  determination  of  the 
containing  rectangle.  This  is  done  arithmetically  from  direct  measurement. 
The  rectangle  thus  obtained  may,  frequently,  be  verified  arithmetically 
by  measurement  of  details.  If  a  root-two  rectangle  be  obtained,  for  example, 
i.  <?.,  a  rectangle  whose  ratio  is  some  recognizable  one  connected  with  the  root- 
two  series,  and  the  width  of  the  foot,  lip  or  neck  either  divided  into,  added  to 
or  subtracted  from  this  ratio,  or  divided  into  the  width  or  height  of  the  whole, 
produces  other  ratios  recognizable  as  belonging  to  the  root-two  series,  a  theme 
in  root-two  is  almost  sure  to  be  found.  Usually  root-two  and  root-three  are 
easier  to  recognize  than  themes  in  the  compound  forms/This  is  due  to  the  fact 
that  root-two  and  root-three  do  not  modulate  or  unite  with  other  shapes.  Com- 
paratively, the  synthetic  use  of  symmetry  is  simple;  the  artist,  however,  must 
understand  basic  principles  and  be  familiar  with  simple  geometrical  construc- 
tion or  the  use  of  a  scale.  The  scale  to  use  is  one  with  units  divided  into  tenths 
because  the  ratios  may  be  read  off  directly  as  numbers.  The  technique  of  area 
or  figure  dissection  is  based  upon  the  diagonal  not  only  to  the  major  shape 
but  to  its  composing  elements.  The  relation  of  the  foot  and  lip  of  a  stamnos  of 


Fig.  11.    Stamnos  06.1021. 176,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


DYNAMIC  SYMMETRY  103 

this  chapter,  Fig.  21,  shows  clearly  the  employment  of  two  sub-diagonals.  A 
well-trained  designer  who  understands  his  symmetry  will  work  rapidly,  use  a 
simple  machinery  and  his  key-plan  will  be  unintelligible  to  any  inferior  in  sym- 
metry knowledge  to  himself.  In  most  cases  his  working  plan  will  not  show  more 
than  a  few  diagonals.  Dynamic  symmetry  produces  in  a  design  the  correlation 
of  part  to  whole  observable  in  either  animal  or  vegetable  growth.  It  is  a  satis- 
fying harmony  of  functioning  parts  which  suggests  a  thing  alive  or  a  thing 
which  has  the  possibility  of  life.  Design  without  an  understood  symmetry  is 
the  negation  of  this. 

Stamnos  06.1021. 176,  Metropolitan  Museum,  New  York,  Fig.  21,  is  a  simple 
square  and  the  elements  of  the  vase  are  proportioned  by  the  dynamic  sub- 
division of  the  containing  shape. 

AB  is  a  rectangle  of  the  whirling  squares.  AC  is  a  diagonal  to  one-half  this 
shape.  It  cuts  the  diagonal  of  the  whole  at  D,  which  point  determines  the 
width  of  the  foot.  This  foot  width  is  equal  to  one-third  of  a  side  of  the  en- 
compassing square.  AP  is  a  .382  rectangle  and  AF  is  the  diagonal  of  half  this 
shape  and  it  intersects  the  diagonal  of  the  whirling  square  rectangle  AE  at  G.  It 


Fig.  22.    Kalpis  06.1021. 192,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


104  DYNAMIC  SYMMETRY 

also  cuts  the  diagonal  of  the  square  HI  at  J.  The  line  GM  cuts  the  diagonal  of 
the  whirling  square  rectangle  KL  to  determine  the  line  NO,  which  fixes  the 
width  of  the  bowl.  The  line  NO  meets  the  diagonal  of  the  whirling  square  rec- 
tangle AB  at  O  and  the  diagonal  of  the  .382  shape  at  N.  This  shows  that  the 
lip  and  the  foot  of  the  vessel  are  proportioned  respectively  in  terms  of  the 
two  main  divisions  of  the  overall  shape,  i.  e.,  .618  and  .382,  and  that  both  foot 
and  lip  are  directly  proportioned  to  the  width  of  the  bowl.  The  theme  is  an 
arrangement  in  diagonals  of  the  two  main  subdivisions  of  the  containing  square 
and  diagonals  of  half  these  shapes. 

Kalpis  06. 102 1. 1 92  in  the  New  York  Museum,  Fig.  22,  is  contained  in  a 
square.  A  small  error  is  shown  at  the  points  where  the  handles  do  not  quite 
touch  the  sides  of  this  square.  The  width  of  the  bowl  and  the  height  define  a 
1.2764  rectangle.  The  fraction  .2764  is  the  reciprocal  of  3.618,  i.  e.,  two 
squares  plus  a  whirling  square  rectangle.  The  area  of  the  lip  and  neck  is  com- 
posed of  these  two  squares,  while  AC  and  DE,  added,  form  the  whirling  square 
rectangle.  AB  is  a  square.  The  width  of  the  foot  is  the  side  of  the  2.618  shape 
FG.  FH  is  a  square,  and  HK  is  1.618.  FI  is  a  whirling  square  rectangle.  The  area 
of  the  foot  elevation  is  composed  of  two  whirling  square  rectangles  plus  a  square 
or  the  ratio  4.236. 


CHAPTER  NINE:  SKYPHOI 


7  ^vURING  the  entire  classical  period,  Greek  designers  seem  to  have 
been  searching  for  certain  ideal  shapes  for  certain  purposes. 
The  large  drinking  bowls,  which  we  recognize  under  the  general 
name  of  Skyphoi,  in  their  general  proportions,  illustrate  this. 
The  overall  shape  scheme  of  these  vases  approximates  a  ratio 


of  one  and  three-quarters.  Modern  designers  would  frankly  accept  this  ratio 
and  not  trouble  themselves  about  subtle  refinements  on  either  the  plus  or  minus 
side  of  so  obvious,  and  consequently  commonplace,  an  area. 

The  employment  of  ratios  either  a  little  less  or  a  little  more,  than  one  to  one 
and  three-quarters,  suggests  conscious  effort  to  get  away  from  an  ordinary  rec- 
tangle. Again,  the  skyphoi  shapes  curiously  parallel  the  Nolan  amphorae 
forms,  the  difference  of  the  outstanding  or  containing  rectangle  in  most  cases 
being  simply  that  of  position.  The  sides  of  the  skyphoi  rectangles  rest  horizon- 
tal, the  sides  of  the  amphorae  shapes,  perpendicular.  Also,  Greek  classic 
artists  wasted  little  design  material.  This  is  shown  by  their  use  of  curves. 
Practically  all  convex  curves  of  one  design  are  repeated  as  concave  curves  in 
other  creations.  For  example,  the  convex  curve  of  the  pelike  is  the  concave 
curve  of  the  pyxis.  The  convex  curves  of  the  lekythos  are  the  concave  curves 
of  the  calyx  krater. 

Convex  cups  have  their  concave  counterparts,  a  sort  of  reverse  echo  in  forms 
which  may  be  termed  an  inversion  of  a  theme. 


Fig.  I.    Black-figured  Skyphos  in  the  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


io6 


DYNAMIC  SYMMETRY 


The  two  skyphoi,  Figs,  i  and  4  of  this  chapter,  should  be  compared.  The  ele- 
vation of  each  shows  the  same  rectangle.  One  vase  is  in  the  Metropolitan 
Museum,  New  York,  the  other  in  the  Museum  of  Fine  Arts,  Boston.  The  rec- 
tangle was  a  favorite  as  it  appears  repeatedly. 


Fig.  2. 


Fig-  3- 


The  early  black-figured  skyphos  in  the  New  York  Museum,  06.1021.49,  Fig. 
1,  has  an  overall  ratio  of  1.854,  three  whirling  square  rectangles,  .618  x  3,  AC, 
CD  and  DB.  The  bowl  ratio,  however,  is  1.382,  GI.  By  construction,  as  shown 
by  the  line  GH  and  the  area  HB,  it  will  be  noticed  that  the  general  scheme  is 
that  of  two  overlapping  whirling  square  rectangles,  IB  and  AG,  the  overlap 
being  the  1.382  shape  in  the  middle. 

Fig.  2  shows  the  three  whirling  square  rectangles.  Fig.  3  shows  the  overlapping 
whirling  square  rectangles.  A  1.382  shape  divided  by  two  equals  two  square 
and  root-five  areas,  IK,  KH,  and  I  J,  JH  in  Fig.  1  are  squares,  MK,  KL  two  root- 


■)  — ' — i — ^  1  

 f 

.  b.'  .';■..] 

 L  i 

v   1  i 

f 

( •  1 

==).. 

Fig.  4.  Adams  Skyphos  in  the  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


DYNAMIC  SYMMETRY  107 

five  areas.  The  width  of  the  foot  is  determined  by  the  point  R  and  SL  is  a 
square.  OP  and  OA  are  squares.  Thus  the  vase  without  its  foot  would  be  a 
root-four  area.  HB  is  composed  of  two  whirling  square  rectangles  plus  a  square. 

A  black-figured  skyphos  loaned  to  the  Boston  Museum  by  the  late  Henry 
Adams,  Fig.  4,  has  the  same  shape  as  No.  06.1021.49  in  the  Metropolitan 
Museum,  New  York.  The  overall  ratio  of  1.854  (.618  multiplied  by  three)  is 
divided  in  exactly  the  same  manner  as  is  the  one  from  New  York.  The  Adams 
vase  has  a  slightly  narrower  foot  as  shown  by  the  point  A,  the  center  of  the 
small  square  BC.  The  bowl  is  1.382  and  the  vase  minus  the  foot  equals  two 
squares  as  shown  by  the  line  DE,  the  diagonal  to  a  square.  FE  and  GH  are 
two  whirling  square  rectangles  overlapping  to  the  extent  of  GI,  the  1.382 
shape.  Dr.  Caskey  has  suggested  the  sequence  of  subdivision  in  the  three  small 
diagrams,  Figs.  5,  6  and  7.  The  picture  on  this  vase  shows  clearly  that  the 
Greek  artist  at  the  time  was  incomparably  better  as  a  designer  than  as  a  figure 
draughtsman.  The  figures  of  the  men  riding  the  dolphins  are  crudely  suggested, 
but  the  picture  as  a  design  composition  is  superb. 


vy  ■ 

•vy-' 

A  5 

5  ■/ 

S 

S 

w 

w 

r'5.  

 r.s' 

Fig.  5.  Fig.  6. 


w  '-, 

W 

W 

Fig.  7. 


A  small  black-glaze  skyphos  at  Yale,  Fig.  8,  has  an  overall  ratio  of  2.1213 
or  three  root-two  rectangles,  .7071  x  3  =2.1213  (compare  Kylix,  Fig.  15,  Chap- 
ter IV),  AB,  BC,  CD  are  root-two  rectangles.  AE,  BF  are  squares.  These  squares 
divide  the  area  AB  into  three  squares  and  three  root-two  rectangles.  The  gen- 
eral proportions  are  all  obtained  by  this  subdivision  of  the  root-two  shape  AB. 


io8  DYNAMIC  SYMMETRY 


Fig.  8.    Black-glaze  Skyphos  at  Yale. 
(Measured  and  curves  traced  by  Prof.  P.  V.  C.  Baur.) 


Fig.  9.    Skyphos  76.49,  Boston. 
(Drawn,  measured  and  analyzed  by  L.  D.  Caskey.) 


Red-figured  Skyphos  76.49,  Boston,  Fig.  9,  furnishes  an  overall  rectangle 
with  a  ratio  of  1.8944.  This  is  a  square  plus  two  root-five  rectangles.  The  eleva- 
tion of  the  bowl  however  is  1.236,  or  two  whirling  square  rectangles,  and  the 
logical  subdivision  of  one  of  these  determines  the  proportionate  relation  of  the 
details  of  foot  and  decorative  bands.  The  points  C,  D  and  E  in  the  rectangle  AB 
are  clear. 


DYNAMIC  SYMMETRY 


Fig.  10.    Boston  Skyphos  13.186. 


Skyphos  13.186  in  the  Boston  Museum,  Fig.  10,  has  a  bowl  ratio  of  1.309 
and  an  overall  area  of  1.809.  The  whirling  square  rectangle  AB  is  derived 
from  the  overall  shape.  The  center  of  the  square  DE  fixes  the  width  of  the 
bowl.  The  relation  of  the  bowl  to  the  meander  band  beneath  the  picture  is 
shown  by  C  and  F.  The  points  G  H  show  that  the  meander  band  at  the  top 
of  the  picture  is  related  to  the  foot. 


Fig.  11.    Boston  Skyphos  01.8076. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


no  DYNAMIC  SYMMETRY 

Skyphos  01.8076  in  the  Boston  Museum,  Fig.  11,  has  a  bowl  ratio  of  1.236 
and  an  overall  area  of  1.764.  This  latter  ratio  frequently  appears  in  Greek 
design. 


Fig.  12.    Boston  Skyphos  01.8032. 


Skyphos  01.8032  in  the  Boston  Museum,  Fig.  12,  has  a  bowl  ratio  of  1.236, 
and,  apparently,  an  overall  area  which  is  a  root-three  rectangle.  This  is  the 
only  case  in  over  four  hundred  examples  of  Greek  design  where  a  root-three 
figure  was  apparently  used  in  connection  with  a  whirling  square  rectangle. 


Fig.  13.    Yale  Skyphos  398 


Black-glaze  Skyphos  398,  at  Yale,  Fig.  13,  has  a  bowl  ratio  of  1.236  and 
an  overall  area  of  1.809.  AD  is  a  whirling  square  rectangle  from  the  1.809  area- 
GB  is  the  diagonal  to  a  square  and  the  point  H  shows  that  without  its  foot 
the  vase  is  a  root-four  area. 


DYNAMIC  SYMMETRY 


1 1 1 


Fig.  14.    Skyphos  06.1079,  Metropolitan  Museum,  New  York. 


A  skyphos  from  the  New  York  Museum,  Fig.  14,  has  a  bowl  ratio  of  1.236 
and  an  overall  area  of  1.809.  AB  is  a  whirling  square  shape  from  the  bowl 
while  EC  is  a  similar  figure  from  the  1.809  ratio. 


Fig.  15.    Boston  Skyphos  10.176. 


Skyphos  10.176  in  the  Boston  Museum,  Fig.  15,  has  a  bowl  ratio  of  1.236, 
while  over  all  it  is  1.809.  The  picture  composition  is  placed  within  the  square 
AB. 


112  DYNAMIC  SYMMETRY 


Fig.  1 6.    Yale  Skyphos  397. 


Black-glaze  Skyphos  397  in  the  Stoddard  Collection  at  Yale,  Fig.  16, 
has  a  1.854  ratio.  As  AB  is  the  diagonal  to  a  square  the  area  of  this  vase  with- 
out the  foot  is  equal  to  a  root-four  rectangle.  The  bowl,  as  shown  by  E,  has  a 
1.236  ratio  or  two  whirling  square  rectangles.  G  is  the  center  of  the  square 
DC.  (See  Figs.  1  and  4,  this  chapter.) 


Yale  black-glaze  Skyphos  399,  Fig.  17,  has  an  overall  ratio  of  1.854  while 
the  bowl  is  1.236,  and  the  vase  without  the  foot  is  a  root-four  rectangle. 


A  black-glaze  skyphos  in  the  Stoddard  Collection  at  Yale,  400,  Fig.  18,  has 
a  ratio  of  1.854  or  three  whirling  square  rectangles.  The  bowl  ratio  is  1.236  or 
two  whirling  square  rectangles.  AC,  CG  are  two  squares.  The  point  H  is  the 
intersection  of  the  diagonals  of  the  square  HJ  and  the  two  whirling  square 
rectangles  AI. 


CHAPTER  TEN:  KYLIKES 


r 


"^HE  adjustment  of  the  handles  on  a  kylix  to  maintain  a  pro- 
portional relationship  with  the  bowl  and  minor  elements  of  the 
design  seems  to  have  been  a  difficult  technical  problem  to  the 
Greek  potter.  The  great  width  of  the  bowl  compared  to  its 
height  and  the  delicacy  of  both  stem  and  bowl  supplies  an 
uncertain  foundation  for  the  attachment  of  the  two,  comparatively,  heavy 
handles.  When  the  kylix  was  first  submitted  to  analysis  the  varying  height  of 
the  handles  suggested  that  the  pottery  designers  had  frankly  met  the  difficulty 
of  adjustment  by  making  allowance  for  an  error.  This  was  found  to  be  true 
because,  while  the  handles  were  sometimes  high  and  sometimes  low,  there  was 
one  feature  of  this  arrangement  which  was  practically  stable.  This  was  their 
width  in  relation  to  the  bowl.  The  makers  of  the  kylix,  therefore,  must  have 
raised  or  lowered  the  handles,  after  they  were  attached  and  while  the  clay 
was  still  workable,  so  the  width  should  remain  true. 

Of  course,  the  handles  of  the  kylix  may  be  ignored,  as  they  may  also  be  in  the 
skyphoi,  and  the  analysis  confined  to  the  bowl,  foot  and  other  details;  but 
the  Greek,  apparently,  did  not  ignore  the  handle  adjustment  in  any  type  of 
pottery  when  they  extended  beyond  the  rectangle  of  the  bowl,  a  fact  clearly 
shown  by  the  amphorae.  In  this  vase  class  there  are  many  examples  with  han- 
dles both  inside  and  outside  the  bowl  rectangle;  when  outside  they  are  almost 
invariably  finely  worked  and  highly  finished,  when  inside  the  reverse  occurs. 
The  Greek  pottery  collection  in  the  Boston  Museum  of  Fine  Arts  is  unusually 
rich  in  kylikes  and  Dr.  Caskey  has  given  them  careful  attention,  as  the  table 
in  this  chapter  shows.  This  table  contains  seventeen  examples  of  red-figured 
kylikes  completely  examined.  The  complete  list  comprises  fifty-four  examples. 

This  table  is  interesting.  First  it  shows  that  five  out  of  the  seventeen  are 
themes  in  root-two  while  the  other  twelve  are  design  arrangements  in  the  com- 
pound figures  derived  from  the  proportions  found  in  the  dodecahedron  or  the 
icosahedron.  The  relation  of  the  details  to  the  overall  shape  as  shown  in  the 
classification  is  striking.  Of  the  seventeen  there  are  six  where  the  width  of  the 
foot  is  equal  to  the  height  of  the  bowl,  or  one  side  of  a  square  in  the  overall 
shape.  The  reader  will  recognize  the  tabled  ratios  as  representing  dynamic 
areas  which  have  appeared  frequently  in  the  vases  so  far  described. 

In  every  example  the  details,  as  sub-ratios,  show  a  recognizable  theme  in  terms 
of  the  overall  shape.  Of  the  root-two  shapes  there  are  three  overall  ratios  of 
3.4I42,  or  two  squares  plus  a  root-two  rectangle. 


A 
O 
H 
cn 
O 
CQ 

oT 
H 

< 

i— i 
fa 

fa 

O 


D 
fa 

P 


fa 

fa 
fa 

Q 

fa 

O 

O 
i— i 
fa 

u 
< 

fa 
PQ 


o 

<U 
> 

O 

o 


C 


bo 
C 


£ 


DYNAMIC  SYMMETRY  115 


CASKEY'S  TABLE  OF  R.  F.  KYLIKES 


Museum  No. 

Overall  Shape 

Bowl 

Foot 

Stem 

Base  of  Stem* 

Projection  of 
Handles 

89.272 

3.000 

2.382 

.8944 

•309 

95-35 

3.090 

2.472 

.8944 

.236 

.6552 

•309 

01.8074 

3.090 

2.236 

1. 000 

■309 

•427 

95-32 

3-236 

2.618 

1. 000 

.691 

•309 

00.338 

3-236 

2.528 

1. 000 

.236 

.764 

•354 

01.8020 

3-236 

2.528 

1. 146 

.236 

.691 

•354 

01.8022 

3-236 

2.618 

•927 

•764 

•309  - 

10.195 

3-236 

2.618 

1. 000 

■236 

.618 

•3°9 

8Q.270 

3-382 

2.618 

1 .000 

•309 

.  C4.C 

.382 

1353-15 

3-382 

2.764 

1. 000 

.764 

•3°9 

01.8038 

3-528 

2.764 

1.09 

•764 

.382 

01.8089 

3-854 

2.854 

1-545 

ROOT-TWO  SHAPES 

3.0606 

2-3535 

•9393 

■3535 

T  CI  C 

95-33 

3-4H2 

2.4714 

1.0672 

.4714 

•47 1 4 

98-933 

2.7071 

1. 000 

00.345 

3-4H2 

2.7071 

1 .0606 

.7071 

•3535 

13.82 

3-4H2 

2.7071 

.2929 

•3535 

The  overall  shape  of  the  early  black-figured  kylix,  03.784  in  the  Boston  Mu- 
seum, Fig.  1,  is  represented  by  the  ratio  2.854.  The  bowl  ratio  is  a  root-five  rec- 
tangle. The  width  of  the  foot  is  a  side  of  the  square  in  a  root-five  shape.  The 
difference  between  the  square  root  of  five,  2.236,  and  the  ratio  2.854  is  .618; 
consequently,  the  handles,  as  represented  by  AE  and  DF,  are  each  equal  to  two 
whirling  square  rectangles.  The  bowl  fills  two  whirling  square  rectangles  as 
shown  by  AG,  GD,  and  the  area  of  which  the  foot  is  a  side  is  composed  also  of 
two  such  shapes  as  shown  by  CG  and  GB.  The  scheme  of  the  kylix,  therefore, 
is  a  theme  throughout  in  double  whirling  square  rectangles. 

*Base  of  stem  is  the  slightly  raised  ring  on  top  of  the  foot. 


n6  DYNAMIC  SYMMETRY 


509  — 1 —  .fe  1  e  — I —  1   1 —  -616  — |—  .509 


Fig.  I.    Black-figured  Kylix  03.784  in  the  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


Fig.  1.    Boston  Eye  Kylix  13.83. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 

A  large  Boston  eye  kylix,  Fig.  2,  is  a  theme  in  root-two.  The  overall  area  ratio  is 
3.0606.  The  bowl  area  is  2.3535.  The  two  handle  areas,  added,  represent  .7071, 
the  reciprocal  of  root-two,  and  therefore  a  root-two  shape.  Each  handle  area 
must  then  be  composed  of  two  root-two  areas.  The  bowl  area,  ls  com- 

posed of  two  squares  plus  .7071  divided  by  two,  or  two  plus  .3535.  BE,  FC  are 
the  squares  and  FG  is  the  area  composed  of  two  root-two  figures.  The  areas 
HI  and  JK  are  each  a  root-two  rectangle  and  JF  is  the  difference  between  .7071 
and  unity  or  .2929. 


DYNAMIC  SYMMETRY 


117 


\  — **  f  I— 

 *  7 

\                         /                                                        /  | 

•._J. 

-3»S3           .  .555 

SOS                                 »               .553  » 

Fig.  3.    Yale  Kylix  167. 


A  heavy  red-figured  kylix  at  Yale,  Fig.  3,  has  an  overall  area  ratio  of  2.618. 
The  bowl  ratio  is  1.927,  the  fraction  being  .618  plus  .309.  The  width  of  the  foot 
is  the  end  of  an  .809  shape.  The  major  area  is  divided  curiously.  The  total  area 
of  the  handles  gives  a  .691  shape,  one-half  of  which  is  -3455-  The  area  AO,  there- 
fore, is  a  square  and  a  root-five;  AP  is  also  such  a  figure,  consequently  it  is  the 
reciprocal  of  AO,  and  the  diagonals  to  both  shapes  meet  at  right  angles  at  Q. 
EF  is  composed  of  four  root-five  rectangles.  FG  equals  two  whirling  square 
rectangles;  AH  and  ID  are  square  plus  root-five  shapes.  The  points  J,  K,  L,  M, 
N  are  clear. 


Fig.  4.    Kylix  92.2654,  Boston. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


n8 


DYNAMIC  SYMMETRY 


Kylix  92.2654  at  Boston,  Fig.  4,  has  an  overall  ratio  of  1.882,  the  bowl 
1.382.  This  leaves  for  the  handles  .5  or  two  squares.  When  .5  is  divided  by 
two  it  will  be  noticed  that  the  space  on  each  end  in  excess  of  the  bowl  is 
composed  of  four  squares.  The  1.382  rectangle  divided  by  two  furnishes  two 
.691  rectangles,  each  of  which  is  composed  of  a  square  plus  a  root-five  rec- 
tangle. The  relation  of  the  foot  to  the  bowl  is  shown  by  the  intersection  of  diago- 
nals to  two  squares  and  the  two  .691  forms. 

The  area  AB,  which  is  determined  by  the  line  formed  by  the  juncture  of  the 
lip  with  the  bowl,  supplies  the  ratio  1.7236,  i.  e.,  a  square  plus  a  1.382  shape, 
.7236  being  the  reciprocal  of  1.382.  CB  is  this  form  and  it  is  divided  into  two 
.691  shapes  by  the  line  DE. 


w 

U 

Fig.  5.    New  York  Kylix  by  Nikosthenes. 
(Measured  and  drawn  by  the  Museum  Staff.) 


A  large  eye  kylix  in  the  New  York  Museum,  14.136,  Fig.  5,  signed  by  Ni- 
kosthenes, has  an  overall  area  of  three  squares.  The  bowl  area  however  is 
2.4472,  /'.  e.,  two  squares  plus  root-five.  The  width  of  the  foot  in  relation  to  the 
height  is  .9472,  which  is  root-five,  .4472  plus  .5  or  two  squares,  or  1 .4472,  a  square 
plus  root-five,  minus  .5  or  two  squares.  The  foot  area  AB  is  composed  of  two 
squares,  and  CD  is  one  square.  The  areas  EF,  BG  are  each  one  and  one-third. 
The  areas  EH  and  GI  are  each  composed  of  two  squares  plus  a  whirling 
'  square  rectangle.  There  is  much  evidence  in  this  vase  that  the  designer  had 
been  trained  in  static  symmetry.  The  method  of  arranging  the  units  of  form 
have  a  distinct  static  flavor. 

A  large  red-figured  kylix,  06.1021. 167  in  the  New  York  Museum,  Fig.  6, 
supplies  an  overall  ratio  of  three  squares.  The  width  of  the  bowl  in  relation  to 
the  height  however  is  2.4142,  i.  e.,  a  root-two  rectangle  plus  a  square.  The 
two  root-two  rectangles  AB,CD  have  ends  equal  in  length  to  half  the  diagonal 
of  one  of  the  major  squares. 


DYNAMIC  SYMMETRY  119 


Fig.  6.    Kylix  06.1021. 167,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


Fig.  7.    Boston  Kylix  95.35. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


A  large  kylix,  95.35  in  the  Boston  Museum,  Fig.  7,  has  an  overall  area  of  3.090 
or  five  whirling  square  rectangles,  .618x5  =3.090.  The  bowl  area  is  four  whirling 
square  rectangles  or  2.472.  This  latter  fraction  subtracted  from  3.090  equals 
.618,  therefore  the  handle  areas  are  each  composed  of  two  whirling  square 
rectangles.  In  the  whirling  square  rectangle  BC  the  line  representing  the  width 
of  the  foot  passes  through  the  point  D.  Therefore  the  foot  width  is  equal  to 
the  end  of  an  area  represented  by  two  root-five  rectangles.  AB  is  one  of  these. 

The  overall  ratio  of  the  black-figured  kylix,  06.1097  in  the  Metropolitan 
Museum,  New  York,  Fig.  8,  is  2.472  or  .618  multiplied  by  4.  The  bowl  ratio  is 
1.854  or  .618  multiplied  by  3.  AB  is  the  major  square  in  the  reciprocal  BC  of 
the  whirling  square  rectangle  BD. 


i2o  DYNAMIC  SYMMETRY 


Fig.  8.    Black-figured  Kylix  06.1097,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 

Ring-foot  Kylix  01.8089,  Museum  of  Fine  Arts,  Boston,  Figs,  ga  and  gb. 
Overall  ratio  3.854,  bowl,  2.854.  Three  whirling  square  rectangle  reciprocals, 
.618,  multiplied  by  three,  equal  1.854,  a  common  shape  in  Greek  design,  espe- 
cially among  the  skyphoi.  The  ratios  3.854  and  2.854  are  apparent.  In  one  case 
it  is  1.854  plus  two  squares,  the  other  1.854  plus  one  square. 


.5  -X—  .6545  — H —  1.545  — .6545  — X—  .5 


M  (r,  A  oi.«o»<j 

Fig.  9«. 


Fig.  gb. 

Boston  Kylix  01.8089. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


DYNAMIC  SYMMETRY 


121 


The  width  of  the  foot  is  the  side  of  a  rectangle  composed  of  two  whirling 
square  rectangles  and  a  half.  .618  multiplied  by  two  and  a  half  equals  1.545. 
The  area  between  the  handles  and  bowl,  on  each  side,  equals  two  squares  or  .5. 
The  area  between  the  foot  and  the  bowl,  on  each  side,  is  composed  of  a  square 
plus  two  whirling  square  rectangles  divided  by  two,  1.309  divided  by  two 
equals  .6545  and  1.545  plus  1.309  equals  2.854.  Another  arrangement,  as  in  by 
makes  clear  the  relationship  of  detail  in  the  design. 


A 


Shift       3  2  36  -   I  w  I  W  I 
Bowl      2  613   -   I  S  |w|  5  [ 

Fig.  10. 

(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


Kylix  01.8022,  Museum  of  Fine  Arts,  Boston,  Fig.  10,  has  an  overall  shape 
of  two  whirling  square  rectangles  or  3.236,  while  the  bowl  proportion  is  a  whirl- 
ing square  rectangle  plus  a  square,  or  2.618. 


M.F.  A.    01  B057 
Alt.c  f;j„V«i    K,l.«  3. 134    -  1-1-1 

Fig.  11. 

(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


122 


DYNAMIC  SYMMETRY 


AB  is  a  whirling  square  rectangle  as  are  also  AC  and  DE  and  EF  and  FG  and 
GH,  the  detail  proportions  being  simply  that  of  continued  reciprocals.  The 
areas  of  the  handles  are  each  two  whirling  square  rectangles. 

Black-figured  Eye  Kylix  01.8057. in  the  Boston  Museum,  Fig.  11,  has  an 
overall areaof  3. 236,  two  whirling  square  rectangles.  The  bowl  area  is  2.618.  The 
difference  between  2.618  and  3.236  is  .618,  therefore  the  handle  areas,  AB  and 
CD,  are  each  composed  of  two  whirling  square  rectangles.  The  ratio  2.618  is  a 
whirling  square  rectangle,  1 .618,  plus  a  square.  The  width  of  the  foot  is  the  side 
of  this  square,  i.  e.,  the  width  of  the  foot  is  equal  to  the  height  of  the  bowl. 

Yale  Kylix  165,  Fig.  12,  has  an  overall  ratio  of  3.236.  AB,  CD  are  whirling 
square  rectangles.  E  is  the  intersection  of  a  whirling  square  diagonal  with  a 
diagonal  of  the  whole.  The  points  F,  G,  H  and  I  are  clear. 


Fig.  12. 


Fig.  I.  Kantharos  in  the  Boston  Museum. 
(Measured  and  drawn  by  L.  D.  Caskey.) 


CHAPTER  ELEVEN:  VASE 
ANALYSES,  CONTINUED 

<A  BLACK-GLAZE  Kantharos  at  Boston,  Fig.  i,  has  an  overallratio  of  a 
/\\  square  plus  a  root-five  rectangle  or  the  ratio  1.4472.  AB  and 
/ \  \       CD  are  the  squares  applied  to  either  end  of  the  rectangle  and 

J  ^\  \  their  diagonals  intersect  at  E;  consequently,  the  area  AF  is  com- 
)  V  posed  of  two  squares.  GH,  the  rectangle  of  the  bowl,  is  a  1.309 
area,  GI  is  one-fourth  of  this,  therefore  a  similar  shape  composed  of  a  square 
GO  and  two  whirling  square  rectangles  IN.  The  square  GO  is  divided  into  the 
squares  JN,  KN,  LN  and  MN.  The  point  R  is  the  center  of  the  square  PQ. 

The  Greek  Olpe  or  Jug  07.286.34,  Metropolitan  Museum,  New  York,  Fig.  2, 
is  a  design  within  the  rectangle  1.9045.  The  fraction  .9045  multiplied  by  two 
equals  1.809.  The  relation  of  this  ratio  to  the  whirling  square  rectangle  and  the 
subdivisions  of  the  square  made  by  the  pentagon,  is  apparent  (see  Chapter 
III).  The  handle  of  the  olpe  extends  beyond  the  rectangle  made  by  the  bowl 
far  enough  to  produce  an  overall  ratio  of  1.691.  The  width  of  the  lip  with  the 
full  height  of  the  jug  supplies  a  2.618  shape.  The  width  of  the  foot  with  the 
height  supplies  2.8944,  i.  e.,  two  squares  and  two  root-five  rectangles.  The  area 
AB  is  .691,  a  square  and  root-five  shape.  The  relations  of  the  subdivisions 
of  the  whirling  square  rectangle  AC  are  obvious.  The  width  of  the  bowl  at  its 
juncture  with  the  foot,  in  relation  to  the  full  height,  is  3.090  or  five  whirling 
square  shapes.  The  rectangle  obtained  by  the  full  height  and  the  width  of  the 
neck  at  its  narrowest  point,  is  4.618.  AD  is  1.236. 


124  DYNAMIC  SYMMETRY 


Fig.  a.    Olpe  from  the  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 

A  theme  of  root-two  and  two  squares  appears  in  a  Sixth  Century  B.  C.  leky- 
thos,  in  in  the  Stoddard  collection  at  Yale  University,  Fig.  3.  The  vase  shape 
is  two  squares,  AB  and  BC  in  the  drawing.  AD,  the  height  of  the  bowl,  is  a 
root-two  rectangle.  The  area  CD  is  composed  of  the  square  DS  and  the  root- 
two  rectangle  SN.  A  side  of  a  square,  ES,  produced  from  E  to  J,  determines  the 
root-two  rectangle  JSandfixesthejunctureof  theneckwith  the  body.  A  diagonal 
to  the  whole  cuts  a  side  of  a  square  at  G  to  fix  the  proportion  of  the  lip.  It  also 
intersects  the  end  of  a  root-two  rectangle  at  L  to  determine  the  width  of  the 


AN  EARLY  BLACK-FIGURED  LEKYTHOS, 
STODDARD  COLLECTION  AT  YALE 

A  theme  in  root-two  within  two  squares 


Fig.  3.    Lekythos  111  at  Yale. 
(Measured  by  Prof.  P.  V.  C.  Baur  of  Yale  University.) 

foot  at  its  juncture  with  the  bowl.  The  line  VI  is  the  center  of  the  root-two  rec- 
tangle AD.  This  is  the  line  on  which  the  figures  of  the  picture  stand.  O  is  the 
intersection  of  a  diagonal  of  the  whole  with  the  diagonal  to  the  two  squares 
AP.  The  point  U  is  the  intersection  of  the  diagonal  to  two  squares  with  the 
diagonal  to  the  root-two  rectangle  NS.  The  points  H  and  W  are  fixed  by  a  line 
from  C  to  I.  The  point  K  is  on  the  diagonal  to  the  area  CJ. 

The  ratio  of  a  small  white  lekythos,  06.1021. 125  in  the  Metropolitan  Mu- 
seum, New  York,  Fig.  4,  is  2.7071,  which  is  .7071,  the  reciprocal  of  root-two, 


126 


DYNAMIC  SYMMETRY 


plus  two  squares.  The  coordination  of  detail  to  the  whole  shapejs  entirely  by 
diagonals  of  square  and  root-two. 

An  early  black-figured  dinos,  13.205  in  the  Boston  Museum,  Fig.  5,  is  a  static 
example.  The  curve  of  this  vase,  however,  is  interesting  because  it  shows  clearly 
what,  in  the  writer's  opinion,  was  the  Greek  method  of  relating  curves  to  the 
straight  line  and  area  proportion  in  a  work  of  art.  The  dinos  area  is  four 
squares  high  and  five  wide.  The  width  of  the  lip  is  fixed  by  the  point  D,  the 


Fig.  4.    Lekythos  06.1021. 125,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


DYNAMIC  SYMMETRY  127 

intersection  of  the  diagonal  to  two  and  one-half  squares,  AS,  with  the  diagonal 
of  the  square  GH.  EH  and  FG  are  diagonals  to  two  squares,  F  and  E  being 
midway  between  AH  and  AG.  In  the  large  square  IM  the  lines  ML  and  IK  are 
each  diagonals  to  two  squares.  The  point  T  is  the  center  of  the  vase  at  its  base 
and  J  the  middle  of  the  side  of  the  square  IC.  The  curve  touches  FG  at  Q,  HE 
at  P,  the  point  J,  IK  at  0,  LM  at  N  and  the  point  T.  Artists  will  appreciate 
the  quality  possessed  by  a  curve  of  this  character,  where  it  is  perfectly  related 
to  the  composing  elements  of  a  theme  in  design  and  is  not  in  any  way  mathe- 


Fig.  5.  Dinos  13.205,  Boston  Museum. 
(Measured  and  drawn  by  L.  D.  Caskey.) 

matical.  Curves  were,  apparently,  drawn  by  tangents  in  this  .manner  all  through 
the  Greek  classical  period.  Hardly  a  vase,  among  the  hundreds  so  far  examined, 
fails  to  disclose  this  method  of  relating  curve  to  angle,  area- and  line.  The  con- 
structions necessary  to  show  this  have  been  kept  out  purposely  in  other 
examples  to  avoid  confusion.  No  mathematical  curves  have,  so  far,  been  found 
in  Greek  art. 

The  shape  of  a  black-glaze  oinochoe  in  the  Stoddard  collection  at  Yale  Univer- 
sity, Fig.  6,  is  a  1.4472  rectangle,  asquare  plus  root-five.  AB,  CD  are  eachsquares 
and  CB,  AD  are  each  root-five  rectangles.  A  1.4472  rectangle  divided  into  two 
parts  produces  two  1.382  rectangles.  1.4472  divided  by  2  equals  .7236  and  this 
fraction  is  a  reciprocal  of  1.382.  The  lines  GM  and  FL  pass  through  the  center 
of  the  two  1.382  shapes.  These  lines  intersect  diagonals  to  the  two  root-five  rec- 
tangles at  M  and  L,  determining  the  width  of  the  lip  and  foot,  also  the  height  of 
the  neck  as  shown  by  the  square  HI.  The  line  JK shows  that  the  height  of  the 


DYNAMIC  SYMMETRY 


Fig.  6.    A  Black  Oinochoe  at  Yale. 
(Measured  by  Prof.  P.  V.  C.  Baur  of  Yale  University.) 


Fig.  7.    Olpe  from  the  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


A  BLACK  GLAZE  OINOCHOE  FROM  THE  STODDARD 
COLLECTION  AT  YALE 
A  theme  in  square  and  root-five 


DYNAMIC  SYMMETRY  129 

vase  without  the  handle  is  proportioned  to  the  thickness  of  the  foot  by  the  diag- 
onal to  a  1.382  shape,  as  CE,  and  the  diagonal  to  a  root-five  rectangle,  as  AD. 

The  area  of  the  jug,  Fig.  7,  is  a  perfect  whirling  square  rectangle.  The  details 
are  correlated  by  reciprocals  of  the  major  shape. 


Fig.  8.    Amphora  01.8059,  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 

An  early  black-figured  amphora,oi. 8059  in  the  Boston  Museum,  Fig.  8,  in  area, 
is  a  whirling  square  rectangle.  The  width  of  the  lip  is  the  side  of  a  square  in  the 
whirling  square  rectangle  AB.  In  the  whirling  square  rectangle  CD,  the  line 
EF  is  a  diagonal  to  half  that  shape.  G  is  the  intersection  of  EF  with  the  diag- 
onal of  the  square  HI.  The  remainder  of  the  analysis  is  clear. 

Boston  Amphora  10.178,  Fig.  9,  is  a  perfect  whirling  square  rectangle  and  all 
its  details  are  consistently  correlated. 


130  DYNAMIC  SYMMETRY 


Fig.  9.    Boston  Amphora  10.178. 


Lekythos  13.195  in  the  Boston  Museum, Fig.  10,  has  an  overall  ratio  of  two  and 
a  half  and  the  area  is  divided  in  terms  of  this  shape,  consequently  it  is  a  static 
example.  AB  is  a  diagonal  of  the  whole.  It  intersects  the  halfway  division  of 
the  square  10  at  H  and  the  side  of  two  squares  AC  at  D.  The  line  DT  cuts  the 
diagonal  of  the  square  SL  at  M.  A  line  parallel  with  the  base  meets  the  diagonal 
of  the  whole  at  U.  This  fixes  the  foot  width.  The  line  FHO  is  clear.  The  width 
of  the  lip  is  the  side  of  the  square  FG  and  the  entire  lip  is  composed  of  two 
squares.  The  point  N  is  clear,  the  intersection  of  diagonals  of  the  half  and 
whole.  The  three  squares  LW  fix  the  proportions  of  the  foot.  The  essential 
design  idea  in  this  example  is  the  use  of  a  series  of  correlated  elements  obtained 
by  the  diagonal  of  a  rectangle  made  by  two  and  a  half  squares  cutting  the  sides 
of  two  squares.  These  two  squares  are  placed  at  both  top  and  bottom  of  the 
rectangle. 

An  early  black-figured  lekythos,  95.15,  Fig.  II,  has  an  overall  ratio  of  two 
squares  and  the  method  of  subdivision  shows  that  this  is  a  static  shape.  About 
five  per  cent,  even  less,  of  classic  Greek  design  is  static.  The  Greek  designers 


DYNAMIC  SYMMETRY 


131 


Fig.  10.    Lekythos  13.195,  Boston  Museum. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


who  used  static  symmetry  possibly  were  uninitiated  in  the  craft  guilds.  In  the 
square  AB  the  square  CD  is  equal  to  one-fourth  the  AB  area,  and  CE  is  composed 
of  three  squares.  O  is  the  intersection  of  the  diagonals  of  one  and  two  squares.  F 
is  the  intersection  of  the  diagonals  of  one  and  two  squares.  The  area  IJ  with 
its  diagonal  and  its  influence  at  KLM  is  apparent.  N  is  the  intersection  of  the 
diagonals  of  one  and  two  squares. 

An  early  black-figured  lekythos,  06.1021.60,  Metropolitan  Museum,  New 
York,  Fig.  12,  is  a  simple  root-five  rectangle.  There  is  a  slight  error  in  the 
width  as  shown  where  the  containing  rectangle  does  not  touch  the  sides  of  the 


132 


DYNAMIC  SYMMETRY 


vase.  As  all  the  details  of  the  vessel  are  simple  parts  of  a  root-five  figure  there 
can  scarcely  be  a  doubt  but  that  this  rectangle  was  intended.  A  is  the  center 
of  the  square  BC.  F  is  the  intersection  of  the  diagonal  of  the  square  BC  with 
the  diagonal  of  the  whirling  square  rectangle  BD.  E  is  the  intersection  of  the 
line  FE  with  the  diagonal  of  the  whole.  H  is  the  center  of  a  small  square  and 


Fig.  II.    Boston  Lekythos  95.15. 
(Measured,  drawn  and  analyzed  by  L.  D.  Caskey.) 


IJ  is  an  area  composed  of  two  squares.  G  is  an  intersection  of  the  diagonal  of 
the  whole  with  the  diagonal  of  a  square. 

The  large  white  lekythos,  12.229.10,  Metropolitan  Museum,  New  York,  Fig. 
13,  exhibits  the  rectangle  3.2764.  The  fraction  .2764  is  the  reciprocal  of  3.618. 
The  general  area  of  this  rectangle  will  be  understood  if  two  squares  are  sub- 
tracted. 3.2764  minus  2  equals  1.2764,  and  this  remainder  equals  .8944  plus 
.382.  This  latter  fraction,  which  is  composed  of  two  squares  and  a  whirling 
square  rectangle,  furnishes  the  proportional  area  which  defines  the  details  of  the 
lip.  The  fraction  .8944  equals  two  root-five  reciprocals,  .4472  multiplied  by  2. 


DYNAMIC  SYMMETRY  133 

One  of  these  root-five  shapes  fixes  the  details  of  the  foot.  Many  other  arrange- 
ments of  the  encompassing  area  could  be  made.  For  example;  1.2764  is  com- 
posed of  .7236  plus  .5528.  .7236  is  the  reciprocal  of  1.382,  .5528  is  the  reciprocal 
of  1.809,  1.2764  plus  .7236  equals  2.  1.2764  multiplied  by  2  equals  2.5528,  .7236 
plus  2.5528  equals  3.2764.  Such  combinations  of  area  units  as  this  should  prove 


Fig.  12.    Black-figured  Lekythos  06.1021.60,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 

of  the  greatest  value  to  designers.  All  of  these  areas  may  be  readily  determined 
with  a  scale,  and  after  the  forms  are  studied,  fixed  by  construction. 

Lekythos  G.  R.  540  in  the  New  York  Museum,  Fig.  14,  has  a  ratio  of  root- 
eight,  i.  e.y  root-two  multiplied  by  two.  The  proportional  correlation  of  foot  and 
neck  is  by  root-two  rectangles,  diagonals  of  squares  and  diagonals  of  the  whole. 


134 


DYNAMIC  SYMMETRY 


Red-figured  Lekythos  08.258.23,  Metropolitan  Museum,  New  York,  Fig.  15, 
supplies  a  ratio  of  3.236  or  two  whirling  square  rectangles,  1.618  multiplied 
by  2.  The  subdivisions  of  the  whirling  square  reciprocals  at  the  top  and  bot- 


DYNAMIC  SYMMETRY  135 

torn  of  the  enclosing  rectangle,  which  proportion  the  details  of  the  foot  and  the 
lip,  do  not  need  explanation,  beyond  mention  that  AB  is  a  square  in  the  center 
of  CD,  this  area  being  a  whirling  square  rectangle. 

The  red-figured  lekythos,  G.  R.  589,  Metropolitan  Museum,  New  York,  Fig. 
16,  supplies  the  ratio  1.528  (compare  Amphora,  Fig.  1,  page  91,  Chapter  VIII). 
This  form  may  be  subdivided  into  two  1.309  shapes,  1.528  divided  by  two 


F=                                 >  / 

\  / 
G'\ 

 -A  ^  

/ 

c  ' 

:               )  • 

Fig.  14.    Lekythos  G.  R.  540,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


136 


DYNAMIC  SYMMETRY 


Fig.  15.    Red-figured  Lekythos  08.258.23,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


equals  .764,  the  reciprocal  of  1.309,  or  it  may  be  treated  as  a  square  plus  .528. 
This  fraction  is  the  reciprocal  of  1.8944,  i.  <?.,  a  square  plus  two  root-five  rec- 
tangles. Analysis  shows  that  the  second  was  the  method  of  subdivision  used 
by  the  Greek  designer. 

x^B  is  the  major  square  and  BC  the  rectangle  consisting  of  a  square  and  two 
root-five  rectangles.  DE  is  this  secondary  square  and  DC,  EB  the  two  root-five 
shapes.  GH  are  two  points  obtained  by  the  intersection  of  the  diagonal  of  the 
whole  with  the  side  of  the  major  square.  The  general  construction  of  the  lip 


DYNAMIC  SYMMETRY  137 

and  neck  follow  proportional  subdivisions  of  the  secondary  square  and  the  two 
root-five  figures  by  diagonals  to  the  square,  diagonals  to  the  whole  and  diago- 
nals to  a  square  plus  a  root-five  shape.  The  diagonals  of  the  whole  cut  the 
diagonals  of  the  secondary  square  and  a  root-five  figure  at  I  and  J.  These 
points  fix  the  width  of  the  lip.  The  points  K  and  L  are  intersections  of  the  diag- 
onals of  the  secondary  square  with  diagonals  of  a  square  plus  a  root-five  figure. 
M  and  N,  two  points  directly  connected  with  the  proportions  of  the  foot,  are 
centers  of  the  two  root-five  shapes. 


Fig.  16.    Red-figured  Lekythos  G.  R.  589,  Metropolitan  Museum,  New  York. 
(Measured  and  drawn  by  the  Museum  Staff.) 


CHAPTER  TWELVE:  STATIC  SYMMETRY 


^\\HE  basic  idea  in  static  symmetry,  particularly  as  it  is  found 
applied  in  Saracenic  and  in  mediaeval  art,  was  stated  by  the 
writer  in  a  paper  read  before  the  Society  for  the  Promotion 
of  Hellenic  Study  in  London  in  the  autumn  of  1902.  At  that 
time  dynamic  symmetry  had  not  been  formulated  and  it  was 
the  writer's  belief  then  that  the  static  type  would  be  found  in  Greek  art.  Further 
research  proved,  however,  that  this  was  true  only  to  a  small  extent. 

Static  symmetry,  as  found  in  both  nature  and  art,  often,  is  radial.  In  this 
respect  it  is  a  symmetry  of  focus,  an  orderly  distribution  of  shapes  or  com- 
posing units  of  form  about  a  center.  Almost  invariably  these  units  of  form  are 
parts  or  logical  subdivisions  of  the  regular  figures,  the  equilateral  triangle,  the 
square  and  the  regular  pentagon.  The  two  former  predominate.  The  latter  was 
used  generally  as  a  pattern.  Many  Gothic  rose  windows  furnish  examples  of 
pentagonal  pattern.  Static  symmetry  of  a  radial  character  is  regulated  by  a 
binary  or  doubling  ratio,  which  is  inherent  in  the  equilateral  triangle  and  the 
square.  These  two  regular  figures  in  nature  may  result  from  cell  packing.  If  a 


Fig.  1. 


series  of  circles  is  considered,  as  in  Fig.  1,  and  the  centers  joined,  a  network 
of  squares  is  produced.  An  aggregate  of  circles,  which  may  be  considered  as 
representing  spheres,  can  be  placed  in  contact  in  but  two  ways,  either  as  in 
Fig.  1  or  as  in  Fig.  1. 


Fig.  2. 

In  this  arrangement  lines  drawn  through  the  centers  of  the  circles  produce  a 
network  of  equilateral  triangles. 

The  relation  of  the  diameter  of  an  inscribed  to  the  diameter  of  an  escribed 
circle  of  an  equilateral  triangle,  is  one  to  two. 


DYNAMIC  SYMMETRY 


139 


Fig-  3-  Fig.  4. 


AB  in  Fig.  3  is  one-half  CD,  and  if  a  series  of  equilateral  triangles  be  arranged 
as  in  Fig.  4  the  ratio  of  the  diameters  of  the  circles  is  binary  or  doubling.  The 
side  of  an  equilateral  triangle  compared  to  the  radius  of  the  escribing  circle  is 
as  one  is  to  the  square  root  of  three.  Very  often  both  in  nature  and  in  art 
forms,  the  spaces  between  the  expanding  circles  in  an  equilateral  triangle  pat- 
tern, will  be  occupied  by  zones  of  form  which  are  root-three  distances  from 
the  center  of  the  system;  i.  e.,  the  radius  of  a  circle  represented  by  such  a 
zone  of  form  units  would  be  equal  to  the  side  of  an  equilateral  triangle  in- 
scribed in  one  of  the  preceding  binary  circles.  But  the  basic  ratio  in  an 
expanding  system  of  this  type  is  binary. 

The  relation  between  the  diameters  of  circles  inscribing  and  escribing  a 
square  is  as  one  is  to  the  square  root  of  two. 

This  relationship  is  shown  in  Fig.  5.  CD  is  to  AB  as  a  side  is  to  a  diagonal  of  a 
square,  or  unity  to  root  two.  But  CD  is  to  EF  as  one  is  to  two. 


Fig.  5- 


In  a  system  of  squares  expanding  from  a  center,  as  in  Fig.  6,  the  relationship 
of  any  three  consecutive  circles,  by  radius  or  diameter,  is  as  1  :  4/2,  :  2.  The 
equilateral  triangle  produces  the  relationship  1  :  4/3  :  2.  The  square  produces 
the  relationship  1  :  4  2  :  2.  In  each  case  the  basic  ratio  is  binary.  There  is  no 
record  that  this  binary  ratio  was  ever  understood  though  there  is  abundant 
evidence  that  equilateral  triangles  and  squares  were  used  consciously  in  art 
for  the  purpose  of  maintaining  definite  relationship  between  the  parts  and  the 
whole  of  a  composition.  These  simple  figures  form  the  base  of  most  of  the 


14°  DYNAMIC  SYMMETRY 

"systems"  of  proportion  which  have  been  produced.  Indeed,  so  many  of  these 
"systems"  have  appeared  during  the  past  fifty  or  seventy-five  years  that  a  list 
of  even  their  names  would  be  wearisome.* 


Fig.  6. 


The  discovery  of  the  design  value  of  the  regular  figures  of  area  is  spontaneous. 
These  shapes  appear  in  the  decoration,  and  often  in  the  building  construction, 
of  many  peoples.  Apparently  continued  use  of  these  elementary  forms  inevi- 
tably produces  a  system  that  is  eventually  recognizable  as  a  definite  art  product 
of  a  people  or  an  age.  This  is  true  of  Saracenic,  Byzantine,  Norman  or  Gothic 
art.  In  decoration,  especially,  the  themes  are  recognizable  by  inspection.  The 
student  of  symmetry  can  hardly  make  a  mistake  in  following  out  the  pattern 
theme  in  any  style  of  art  where  the  regular  figures  are  used.  In  many  styles  of 
architecture  and  decoration,  other  than  the  Greek  and  the  Egyptian,  root-two 
and  root-three  rectangles  often  may  be  found  but  they  are  always  used  in  the 
static  manner.  In  Greece  or  in  Egypt  they  were  used  in  the  dynamic  manner. 
In  static  symmetry  these  two  rectangles  are  produced  as  logical  divisions  of 
some  regular  figure.  The  central  area  of  a  hexagon,  as  in  Fig.  7,  is  a  root-three 
rectangle. 

The  heavy  line  part  of  Fig.  8  is  a  root-two  rectangle. 

Either  of  these  rectangles  would  be  obtained  in  a  multiplicity  of  ways  from 
the  simple  pattern  forms  made  by  squares  and  equilateral  triangles. 

That  squares  and  equilateral  triangles  are  not  found  oftener  in  Greek  and 
Egyptian  design  is  indeed  remarkable.  The  Choragic  Monument  of  Lysicrates 
is  a  Greek  example  of  a  building  in  which  an  equilateral  triangle  appears  and 

*The  reader  is  referred  to  Gwilt's  "Encyclopedia  of  Architecture,"  section  on  Propor- 
tion, for  a  fairly  complete  list,  with  some  detailed  explanation,  of  these  "systems." 
Also  Leonardo  da  Vinci's  sketch  books,  the  Note  Book  of  Villars  de  Honecourt,  a 
Twelfth  Century  French  architect,  and  the  published  works  of  Viollet  le  Due. 


DYNAMIC  SYMMETRY 


141 


Fig.  7.  Fig.  8. 

the  Tower  of  the  Winds  is  an  example  of  a  square  used  in  a  static  manner  to 
obtain  an  octagon.  Both  of  these  structures  are,  however,  comparatively  late. 
The  equilateral  triangle  appears  often  in  tripod  forms  and  in  chariot  wheels 
of  six  spokes,  but  the  square,  as  a  proportioning  factor  in  decoration  or 
construction,  is  strangely  rare.  The  Greek  and  Egyptian  methods  of  using 
static  symmetry  are  quite  different  from  those  of  other  ages  and  peoples.  In 
the  art  of  the  former  it  is  almost  invariably  employed  as  an  area  in  rectangle 
form,  which  is  subdivided  into  multiple  squares.  For  example,  a  Greek  design 
whose  greatest  width  is  some  even  multiple  of  its  greatest  length,  as  1  :  2,  1  : 
1  :  1.  1/3,  1  :  2^2,  1  :  2.  2/3,  etc.,  is  almost  sure  to  have  its  details  expressi- 
ble in  logical  subdivisions  of  the  containing  shape.  Any  of  the  static  examples 
of  Greek  pottery  shapes  in  this  book  exemplify  the  idea.  The  Greeks,  however, 
seem  always  to  have  been  fond  of  subtleties.  They  seemed  to  enjoy  finding 
hidden  squares.  In  a  shape  composed  of  two  squares,  as  in  Fig.  9,  they  would 


Fig.  9. 

use  the  diagonals  of  the  whole  and  the  diagonals  of  the  half  to  obtain  the 
smaller  square.  Without  the  construction  lines  the  relation  of  the  small  to  the 
two  larger  squares  is  not  obvious.  The  early  black-figured  dinos,  page  127,  is  an 
example  of  the  subtle  use  of  squares  to  obtain,  not  only  structural  but  also 
curve  relationship.  Greek  practice  in  static  symmetry  was  not  essentially 


142  DYNAMIC  SYMMETRY 

different  from  what  it  was  in  dynamic.  The  latter  type  was  simply  a  more 
powerful  andjlexible  instrument. 

The  modern  designer  is  much  at  fault  in  failing  to  realize  that  unless  some 
type  of  symmetry  is  employed  in  aft,  design  does  not  exist.  Indeed,  it  is  ex- 
traordinary that  the  modern  architect  almost  invariably  fails  to  recognize  the 
part  played  by  the  regular  figures  in  Gothic  art.  For  example,  he  seems  to  feel 
that  these  pattern  forms,  which  are  so  manifest,  are  arbitrary  and  were  used 
because  they  facilitated  tracery  and  diaper  arrangement.  As  a  matter  of  fact 
they  are  invariably  the  logical  outgrowth  of  a  fundamental  plan  scheme  which 
permeates  a  structure  or  design  throughout,  thus  producing  that  unity  and 
interrelationship  of  parts  and  whole  which  may  be  compared  to  a  like  con- 
dition in  a  snow  crystal.  The  modern  designer  also  fails  to  realize  that  formal- 
ized art  is  impossible  unless  it  is  schematic.  That  even  realistic  representa- 
tion will  lack  integrity  and  force,  and  become  little  better  than  a  photograph, 
unless  it  is  planned  in  area,  i.  e.y  in  two  dimensions.  It  is  because  of  this  lack 
of  understanding  of  schematic  design  that  no  formalized  animals,  for  example, 
appear  in  art  today,  which  can  in  any  way  be  compared  to  those  of  Egypt, 
Greece  or  the  Middle  Ages.  Indeed,  this  is  the  lesson  that  modern  artists  must 
learn;  that  the  backbone  of  art  is  formalization  and  not  realism.  Art  means 
exactly  what  the  term  implies.  It  is  not  nature,  but  it  must  be  based  on 
nature,  not  upon  the  superficial  skin,  but  upon  structure.  Man  cannot  other- 
wise be  creative,  be  free.  As  long  as  he  copies  nature's  superficialities  he  is  an 
artistic  slave.  No  craftsmen  ever  so  thoroughly  understood  this  as  the  Greeks. 
When  they  used  a  flower  or  a  plant  as  a  design  motive  the  superficial  or  acci- 
dental aspect  of  the  thing  was  eliminated.  They  saw  that  nature  was  tending 
toward  an  ideal,  that  the  principles  at  work  underneath  the  surface  of  natural 
phenomena  were  perfect,  but  that  material  manifestations  of  the  operation 
of  these  principles,  as  exemplified  by  animal  and  vegetable  growth,  owing  to 
vicissitudes  of  circumstance  and  the  length  of  time  necessary  for  development, 
were  seldom  or  never  perfect.  Realization  of  nature's  ideal,  however,  and 
understanding  of  the  significance  of  structural  form  should  enable  the  artist  to 
anticipate  nature,  to  attain  the  ideal  toward  which  she  is  tending,  but  which  she 
can  never  reach.  The  Greek  artist  was  always  virile  in  his  creations,  because 
he  adopted  nature's  ideal.  The  modern  conception  of  art  leads  toward  an 
overstress  of  personality  and  loss  of  vigor. 


APPENDIX:  NOTES 


NOTE  I. 


~^HE  idea  that  so  much  care  should  have  been  taken  to  proportion  such  a  commonplace 
article  as  a  clay  pot,  will  probably  strike  the  average  reader  as  fanciful.  And  it  would 
be  so  if  ordinary  pottery  were  under  consideration.  The  vases  considered  here,  however, 
are  Greek,  and  the  Greek  vase  is  unique.  Nothing  like  it  was  made  before  or  has  been  made  since 
the  classic  period.  Moreover,  in  spite  of  the  fact  that  Greek  ceramics  have  received  the  en- 
thusiastic attention  of  archaeological  and  other  writers  during  the  past  one  hundred  years, 
little  is  known  of  the  subject.  Volumes  have  been  written  about  the  pictures  found  on  Greek 
pottery,  but  the  shape  or  form  of  the  vase,  which  is  of  much  greater  importance,  has  been  al- 
most entirely  neglected.  In  the  light  of  dynamic  symmetry,  in  the  close  analytical  inspection 
of  the  shape  which  this  symmetry  makes  possible,  it  is  clear  that  the  classic  vase  has  survived, 
not  because  of  its  decoration  and  picture,  admirable  as  these  often  are,  but  because  of  the  ex- 
traordinary beauty  of  its  form.  Scholars,  since  the  discovery  of  classic  pottery  in  the  Seventeenth 
Century,  have  advanced  many  strange,  and  sometimes  amusing,  theories  to  explain  this  curious 
and  fascinating  product  of  Greek  design.  The  present  situation  in  regard  to  the  subject  is  summed 
up  by  H.  B.  Walters,  Assistant  Curator  of  Greek  and  Roman  antiquities  in  the  British  Museum, 
who  has  written  a  history  of  pottery:  "Any  day  may  bring  forth  a  new  discovery  which  will 
completely  revolutionize  all  preconceived  theories;  hence  there  is  the  constant  necessity  for 
'being  up  to  date,'  and  for  the  adjustment  of  old  beliefs  to  new."  In  his  introduction  to  the  cata- 
logue of  the  Rebecca  Darlington  Stoddard  Collection  of  Greek  and  Italian  vases  at  Yale  Uni- 
versity, P.  V.  C.  Baur  says:  "To  the  ancient  Greek  the  form  of  the  vase  was  of  vital  importance, 
the  vase  painting  was  usually  of  secondary  importance,  a  fact  made  clear  by  the  great  prepon- 
derance of  signatures  of  potters  over  those  of  painters." 

As  a  matter  of  fact,  Greek  pottery  is  one  of  the  greatest  design  fabrics  ever  created.  It  is  an 
artistic  miracle. 

NOTE  II. 


~^HE  "cording  of  the  temple"  was  a  recognized  process  among  the  Egyptians,  carried 
out  by  professional  rope-stretchers  and  attended  with  ceremonies  somewhat  like  those 
seen  at  our  laying  of  the  corner  stone.1  Lockyer  quotes  several  significant  descriptions 
of  the  process  taken  from  wall  inscriptions  at  Karnak,  Denderah  and  Edfu.  The  Pharaoh  him- 
self was  the  chief  actor  and  he  was  supposed  to  be  assisted  by  a  goddess  called  Sesheta,  "the 
mistress  of  the  laying  of  the  foundation  stone."  These  inscriptions  also  confirm  the  importance 
attached  to  careful  orientation. 

"Arose  the  king,"  says  one,  "attired  in  his  necklace  and  feathered  crown;  and  all  the  world 
followed  him,  and  the  majesty  of  Amenemhat.  The  ker-heb,  chief  priest,  read  the  sacred  text 
during  the  stretching  of  the  measuring  cord  and  the  laying  of  the  foundation  stone  on  the  piece 
of  ground  selected  for  this  temple.  Then  withdrew  his  majesty  Amenemhat;  and  King  Userte- 
sen  wrote  it  down  before  the  people."  Another  inscription  represents  Sesheta  as  addressing 
the  king:  "The  hammer  in  my  hand  was  of  gold,  as  I  struck  the  peg  with  it,  and  thou  wast  with 
me  in  thy  capacity  of  Harpedonapt.  Thy  hand  held  the  spade  during  the  fixing  of  its  four  corners 
with  accuracy  by  the  four  supports  of  heaven."  Two  more  inscriptipns  directly  describe  orienta- 
tion: "The  living  God,  the  magnificent  son  of  Asti,  nourished  by  the  sublime  goddess  in  the 
temple,  the  sovereign  of  the  country,  stretches  the  rope  in  joy,  with  his  glance  toward  the  ak 
of  the  Bull's  Thigh  Constellation,  he  establishes  the  temple-house  of  the  mistress  of  Denderah, 
as  took  place  there  before"  and  the  king  says,  "Looking  to  the  sky  at- the  course  of  the  rising 
stars  and  recognizing  the  ak  of  the  Bull's  Thigh  Constellation,  I  establish  the  corners  of  the 
temple  of  her  majesty."  Finally,  regarding  the  building  of  the  temple  at  Edfu,  Lockyer  remarks; 
"the  king  is  represented  as  speaking  thus: — -'I  have  grasped  the  wooden  peg  and  the  handle 
of  the  club;  I  hold  the  rope  with  Sesheta;  my  glance  follows  the  course  of  the  stars;  my  eye  is  on 
Meschet;  ....  I  establish  the  corners  of  thy  house  of  God.'  And  in  another  place: 
'Sir  Norman  Lockyer,  "The  Dawn  of  Astronomy." 


144 


DYNAMIC  SYMMETRY 


....  'I  have  grasped  the  wooden  peg;  I  hold  the  handle  of  the  club;  I  grasp  the  cord  with 
Sesheta;  I  cast  my  face  toward  the  course  of  the  rising  constellations;  I  let  my  glance  enter 
the  constellation  of  the  Great  Bear;  ....  I  establish  the  four  corners  of  thy  temple."' 
This  laying  out  of  the  plan  was  called  by  the  Egyptians  Put-ser,  which  means  literally  "to 
stretch  a  cord."  Having  obtained  a  North  and  South  line,  says  Ball,1  the  rope  fasteners  found 
an  East  and  West  one  by  an  immemorial  geometrical  method  still  in  use  among  engineers  and 
carpenters.  It  was  known  that  a  triangle  of  which  the  sides  were  respectively  3,  4  and  5  units 
long,  was  necessarily  a  right  triangle.  The  Harpedonapt,  therefore,  took  a  rope  AD  with  knots 
tied  at  B  and  C  so  that  AB  was  equal  to  4,  BC  to  3  and  CD  to  5.  Fastening  BC  with  peg  along 
the  north  and  south  line,  he  then  rotated  BC  and  CD  about  B  and  C  until  the  points  A  and  D 
coincided  to  form  the  vertex  of  a  triangle.  BA  was  then  necessarily  at  right  angles  to  BC. 

Clement  of  Alexandria  quotes  Democritus  as  saying:  "I  have  wandered  over  a  larger  portion 
of  the  earth  than  any  man  of  my  time,  inquiring  about  things  most  remote;  I  have  observed  very 
many  climates  and  lands,  and  have  listened  to  very  many  learned  men;  but  no  one  has  yet  sur- 
passed me  in  the  construction  of  lines  with  demonstration;  no,  not  even  the  Egyptian  Harpedo- 
naptae,  as  they  are  called,  with  whom  I  lived  five  years  in  all,  in  a  foreign  land."  Allman,  p.  80. 

It  is  worthy  of  note  that  about  the  same  time  that  Greek  artists  were  creating  their  stu- 
pendous masterpieces,  and  using  root  rectangles  to  correlate  the  elements  of  their  designs, 
in  far  India  designers  of  another  race  were  using  the  same  idea  in  architecture.  The  Hindus 
actually  worked  out  the  root  rectangles  up  to  root  six.  This  is  as  far  as  the  record  goes.  There 
is  no  indication  that  they  knew  anything  of  the  connection  between  root  five  and  extreme  and 
mean  ratio.  The  Hindu  phraseology  is  suggestive.  The  record  of  the  fact  is  contained  in  the 
Sulvasutras  and  is  published  in  a  book  on  Indian  Mathematics  by  George  Rusby  Kaye  (Cal- 
cutta and  Simla).  Mr.  Kaye  says: 

"The  term  Suhasutra  means  'the  rules  of  the  cord'  and  is  .the  name  given  to  the  supplements 
of  the  Kalpasutras  which  treat  of  the  construction  of  sacrificial  altars.  The  period  in  which  the 
Sulvasutras  were  composed  has  been  variously  fixed  by  various  authors.  Max  Miiller  gives  the 
period  as  lying  between  500  and  200  B.  C:  R.  C.  Dutt  gave  800  B.  C:  Biihler  places  the  origin 
of  the  Apastamba  school  as  probably  somewhere  within  the  last  four  centuries  before  the  Chris- 
tian era,  and  Budhayana,  somewhat  earlier:  Macdonnell  gives  the  limits  as  500  B.  C. 
and  A.  D-  200,  and  so  on.  As  a  matter  of  fact,  the  dates  are  not  known  and  those  suggested  by 
the  different  authorities  must  be  used  with  the  greatest  circumspection.  It  must  also  be  borne 
in  mind  that  the  contents  of  the  Sulvasutras,  as  known  to  us,  are  taken  from  quite  modern 
manuscripts;  and  that  in  matters  of  detail  they  have  probably  been  extensively  edited.  The 
editions  of  Apastamba,  Budhayana  and  Katyayana,  which  have  been  used  for  the  following 
notes,  indeed  differ  from  each  other  to  a  very  considerable  extent." 

Reference  to  the  root  rectangles  are: 

"  'The  chord  stretched  across  a  square  produces  an  area  of  twice  the  size.' 

The  reference  here  is  to  the  diagonal  of  a  square,  probably  as  the  operation  would  be  done  by 
a  "rope  stretcher,"  and,  of  course,  would  be  the  first  step  necessary  for  the  determination  of  a 
root-two  rectangle.  The  square  on  the  diagonal  of  a  square  is  twice  the  area  of  a  square  on  the 
side. 

"  'Take  the  measure  for  the  breadth,  the  diagonal  of  its  square  for  the  length;  the  diagonal 
of  that  oblong  is  the  side  of  a  square  the  area  of  which  is  three  times  the  area  of  the  square.' 

Here  is  described  the  construction  of  a  root-two  rectangle  and  the  use  of  its  diagonal  to  ob- 
tain the  side  of  a  root-three  rectangle.  The  square  described  on  the  side  of  a  root-three  rec- 
tangle is  three  times  the  area  of  the  unit  square.  And  so  on. 

"  'The  diagonal  of  an  oblong  produces  by  itself  both  the  areas  which  the  two  sides  of  the 
oblong  produce  separately. 

"  'This  is  seen  in  those  oblongs  whose  sides  are  three  and  four,  twelve  and  five,  fifteen  and  eight, 
seven  and  twenty-four,  twelve  and  thirty-five,  fifteen  and  thirty-six.'  "  Budhayana  edition. 
Translated  by  Dr.  Thibaut. 

'"Short  History  of  Mathematics." 


DYNAMIC  SYMMETRY 


H5 


This  last  description  refers  to  a  geometrical  construction  which  would  be  equivalent  to  the 
forty-seventh  proposition  of  the  first  book  of  Euclid.  That  is,  that  the  square  on  the  hypotenuse 
is  equal  to  the  squares  on  the  two  legs  of  a  right-angled  triangle.  It  is  noteworthy  that  here  the 
hypotenuse  is  called  the  diagonal  of  an  oblong.  This  would  be  an  artist's  statement  of  the  fact 
enunciated  in  the  forty-seventh  proposition.  A  right-angled  triangle  doesn't  mean  as  much  to 
an  artist  as  does  a  rectangle.  The  former  suggests  incompleteness,  the  latter  means  finish,  an 
ensemble. 

The  second  part  of  the  last  statement  refers  to  the  right-angled  triangles  obtained  by  the 
"rope  stretchers"  when  they  used  the  knotted  rope  to  construct  a  right-angled  triangle  on  the 
ground.  In  Egypt,  as  Cantor  says,  this  operation  of  rope  stretching,  as  is  proven  by  the  bas- 
reliefs,  dates  back  to  a  very  early  period,  possibly  the1  first  dynasty.  This  means  that  rope  stretch- 
ing was  an  established  profession  thousands  of  years  before  there  is  an  historical  reference  to 
the  same  thing  in  either  India  or  Greece. 

The  oblong  whose  sides  are  three  and  four  means  the  celebrated  3,  4,  5  right-angled  triangle 
used  for  temple  cording  for  ages.  Three  and  four  units  on  a  knotted  rope  represent  the  two  sides 
of  a  triangle;  the  hypotenuse  is  five  units,  the  squares  on  the  two  sides  being  three  times  three 
equalling  nine, .and  four  times  four  being  sixteen;  and  nine  plus  sixteen  being  twenty-five,  the 
square  of  five. 

The  sides  of  a  triangle  which  are  composed  of  twelve  and  five  units  will  have  an  hypotenuse 
of  thirteen  units.  12X12  =  144,  5X5=25,  144  plus  25  =  169,  13X13=169. 

Fifteen  and  eight  units  have  an  hypotenuse  of  seventeen  units,  the  sum  of  the  squares  of 
fifteen  and  eight  being  two  hundred  and  eighty-nine,  the  square  of  seventeen,  and  so  on. 

Pythagoras,  one  of  the  Greek  philosophers  who  brought  the  knowledge  of  geometry  from 
Egypt  to  Greece,  has  left  us  a  rule  for  obtaining  these  right-angled  triangles  arithmetically, 
beginning  with  odd  numbers.  Later  Plato  supplied  a  rule  beginning  with  even  numbers.  See 
Allman. 

The  early  development  of  science  in  India  was  apparently  slow  and  was  soon  tainted  with 
looseness  and  inaccuracy.  See  T.  L.  Heath's  "Elements  of  Euclid,"  particularly  his  notes  on 
the  forty-seventh  proposition  of  the  first  book.  This  element  of  inaccuracy  flavors  all  Hindu 
art;  indeed,  degree  of  precision  and  clearness  of  expression  are  hall  marks  for  the  art  of  any 
nation.  Hindu  art,  for  example,  is  much  what  Hindu  science  is;  the  same  may  be  said  of  Greek 
art  and  science. 

NOTE  III. 

P  T]  ^HIS  quotation  from  Vitruvius,  the  Roman  writer  on  architecture,  was  used  by  David 
Ramsey  Hay,  a  Scotch  artist  and  author  of  the  early  part  of  the  nineteenth  century, 

_X  who  wrote  several  books  upon  the  subject  of  symmetry  and  proportion.  Hay's  work  is 
noteworthy  as  he  is  the  only  one  of  the  many  who  have  contributed  theories  to  this  subject 
who  was  attracted  to  the  root  rectangles.  The  idea  was  suggested  to  him  by  a  mathematical 
friend  who  was  conversant  with  the  history  of  Greek  geometry.  Hay,  however,  knew  little  of 
the  properties  of  these  area  figures  and  missed  entirely  the  rectangle  of  the  whirling  squares.  It 
is  remarkable,  however,  that  he  tried  to  obtain  the  design  themes  of  Greek  pottery,  in  spite  of 
the  fact  that  in  his  day  little  was  known  about  the  vase  and  he  did  not  have  the  benefit  of  first- 
hand observation.  This  writer,  however,  made  the  mistake  of  trying  to  bring  design  into  the 
domain  of  music.  In  this  attempt  he  not  only  failed  utterly,  but  became  so  confused  that  his 
contribution,  except  for  its  historical  interest,  is  valueless. 

Modern  research  has  entirely  discredited  Vitruvius.  Not  a  single  Greek  example  has  been 
found  which  bears  out  the  Roman  writer's  theory.  As  a  matter  of  fact,  now  that  we  have  dynamic 
symmetry  as  a  guide,  it  is  clearly  to  be  seen  that  this  writer  gives  us  nothing  but  the  echo  of  a 
tradition  and  his  elaborate  instructions  for  constructing  buildings  in  the  Greek  style  constitute 
nothing  more  than  the  Roman  method  of  using  static  symmetry.  The  Romans  were  either  in- 
tentionally misled  by  the  Greek  artists  and  craftsmen,  or,  blinded  by  conceit,  they  jumped  at 
the  conclusion  that  what  was  meant  by  the  Greek  tradition  that  the  "members  of  the  human 


146 


DYNAMIC  SYMMETRY 


body  were  commensurate  with  the  whole"  was  that  the  length  measurements  were  commen- 
surate. Dynamic  symmetry  now  shows  us  that  not  only  are  the  members  of  a  Greek  statue  of 
the  best  period  commensurate  with  the  whole,  but  that  the  same  is  true  of  the  human  figure. 
But  commensurate  means  commensurate  in  area,  not  in  line.  If  a  statue  is  made  wherein  the 
members  are  commensurate  in  line  a  static  condition  necessarily  results.  See  Note  No.  6. 

NOTE  IV. 


~~ ^HOSE  interested  in  the  latest  contribution  to  Morphology  should  read  "Growth  and 
Form"  by  D'Arcy  W.  Thompson,  Cambridge,  1917.  This  is  an  extraordinarily  well- 
written  book  and  the  author's  treatment  of  the  logarithmic  spiral  in  relation  to  uniform 
growth  is  most  able.  It  may  be  said  in  passing,  however,  that  this  author  has  overstressed  the 
value  of  the  "gnomon"  in  some  respects,  and  understressed  it  in  others.  Professor  Thompson, 
however,  gives  the  best  general  explanation  of  the  proportion  or  logarithmic  spiral  in  relation 
to  growth  phenomena  that  has  yet  appeared. 


NOTE  V. 


~^HE  following  notes  and  bibliography  are  by  Professor  R.  C.  Archibald  of  Brown  University. 
The  writer  feels  that  Professor  Archibald's  contribution  is  both  valuable  and  timely  and  that 
it  will  do  much  to  clear  away  the  mystic,  sentimental  and  impracticable  notions  now  prevalent 


among  artists  and  others  in  relation  to  the  terms  "Golden  Section"  and  "Divine  Section." 

NOTES  ON  THE  LOGARITHMIC  SPIRAL,  GOLDEN  SECTION 
AND  THE  FIBONACCI  SERIES1 

I.    The  Logarithmic  Spiral.2 


~^HE  first  discussions  of  this  spiral  occur  in  letters  written  by  Descartes  to  Mersenne  in 
1638,  and  are  based  upon  the  consideration  of  a  curve  cutting  radii  vectores  (drawn 
from  a  certain  fixed  point  0),  under  a  constant  angle,  </>.3  Descartes  made  the  very  re- 
markable discovery  that  if  B  and  C  are  two  points  on  the  curve  its  length  from  0  to  B  is  to  the 
radius  vector  OB  as  the  length  of  the  curve  from  0  to  C  is  to  OC;4  whence  s  =  ap,b  where  j  is 
the  length  measured  along  the  curve  from  the  pole  to  the  point  (p,0),  and  a  =  sec  </>.6  This  leads 
to  the  polar  equation  (1)  p  —  kecS,  where  k  is  a  constant  and  c  =  cot  <£.  The  pole  0  is  an  asymp- 
totic point.  The  pole  and  any  two  points  on  the  spiral  determine  the  curve;  for  the  bisector  of 


1  Most  of  the  following  notes  appeared  in  The  American  Mathematical  Monthly,  April  and  May,  1918, 
but  extensive  additions,  and  some  corrections,  are  here  introduced. 

2  Historical  sketches  and  some  of  the  properties  of  the  curve  are  given  in  F.  Gomes  Teixeira,  Traite  des 
courbes  speciales  remarquables,  tome  2,  Coi'mbre,  Imprimerie  de  l'universite,  1909,  pp.  76-86,  396-399,  etc.; 
in  G.  Loria,  Spezielle  algebraische  und  transzendente  ebene  Kurven,  Band  2,  2.  Auflage,  Leipzig,  Teubner, 
1 91 1,  pp.  60  fF.;  in  Mathematisches  Worterbu'ch  .  .  .  angefangen  von  G.  S.  Kliigel  .  .  .  fortgesetzt  von  C.  B. 
Mollweide,  Leipzig,  Band  4!,  1823,  pp.  429-440. 

3  The  curve  arises  in  the  discussion  of  a  problem  in  dynamics.  For  references  see  the  next  footnote. 

4  Oeuvres  de  Descartes,  tome  2,  publiees  par  C.  Adam  et  P.  Tannery.  Paris,  Cerf,  1898,  p.  360;  also  pp. 
232-234;  (see  Montucla,  Histoire  des  Mathematiques,  nouvelle  edition,  tome  1,  Paris,  1799,  p.  45).  Cf.  I. 
Barrow,  Lectiones  Geometricae,  Londini,  1670,  p.  124;  or  English  edition  by  J.  M.  Child,  London,  Open 
Court,  1916,  pp.  136-9,  198.  From  the  discussion  and  figure  of  Descartes  it  seems  certain  that  he  had  no 
conception  of  O  as  an  asymptotic  point  of  the  spiral.  This  property  of  the  point  was  remarked  in  a  letter, 
dated  July  6,  1646,  from  Toricelli  to  Robervall  {LTntermediaire  des  mathematiciens,  1900,  vol.  7,  p.  95). 
See  also  G.  Loria,  Atti  delta  accademia  dei  Lincei,  1897,  p.  318. 

5  The  intrinsic  equation  s'"R  =  K  represents  a  logarithmic  spiral  when  m  =  —  I,  a  clothoide  when  m  =  1 , 
a  circle  when  m  =  o,  the  involute  of  a  circle  when  m  =  —  5  and  a  straight  line  when  m  ==  00 .  Haton  de  la 
Goupilliere  remarked,  and  Allegret  proved  (Nouvelles  annates  de  mathematiques,  tome  11  (2),  1872,  p.  163,) 
that  the  logarithmic  spiral  may  be  regarded  also  as  a  particular  case  of  the  spiral  sinusoid. 

6  That  is,  the  length  of  the  arc  measured  from  the  pole  is  equal  to  the  length  of  the  tangent  drawn  at  the 
extremity  of  the  arc  and  terminated  by  the  line  drawn  through  the  pole  perpendicular  to  the  radius  vector, 
that  is,  "the  polar  tangent."    The  logarithmic  spiral  was  the  first  transcendental  curve  to  be  rectified. 


DYNAMIC  SYMMETRY 


H7 


the  angle  made  by  the  radii  vectores  of  the  points  is  a  mean  proportional  between  the  radii.  If 
c  =  1  the  ratio  of  two  radii  vectores  corresponds  to  a  number,  and  the  angle  between  them  to  its 
logarithm;  whence  the  name  of  the  curve. 

The  name  logarithmic  spiral  is  due  to  Jacques  Bernoulli.1  The  spiral  has  been  called  also  the 
geometrical  spiral,2  and  the  proportional  spiral;3  but  more  commonly,  because  of  the  property 
observed  by  Descartes,  the  equiangular  spiral.4 

Bernoulli  (and  Collins  at  an  earlier  date)  noted  the  analogous  generation  of  the  spiral  and  loxo- 
drome  ("loxodromica"),  the  spherical  curve  which  cuts  all  meridians  under  a  constant  angle. 
Credit  for  the  first  discovery  that  the  loxodrome  is  the  stereographic  projection  of  a  loga- 
rithmic spiral  seems  to  be  due  to  Collins.5 

As  the  result  of  Descartes's  letters  distributed  by  Mersenne,  Torricelli  also  studied  the 
logarithmic  spiral.  He  gave  a  definition  which  may  be  immediately  translated  into  equation 
(1),  and  from  it  he  obtained  expressions  for  areas,  and  lengths  of  arcs.  These  results  were 
rediscovered  by  John  Wallis6  and  published  in  1659."  Wallis  states  in  this  connection  that  Sir 
Christopher  Wren  had  written  about  the  logarithmic  spiral  and  arrived  at  similar  results. 

During  1 691 -93  Jacques  Bernoulli  gave  the  following  theorems  among  others:  {a)  Logarithmic 
spirals  defined  by  equations  (1)  for  different  values  of  k  are  equal  and  have  the  same  asymptotic 
point;  {b)  the  evolute  of  a  logarithmic  spiral  is  another  equal  logarithmic  spiral  having  the  same 
asymptotic  point;8  (c)  the  pedal  of  a  logarithmic  spiral  with  respect  to  its  pole  is  an  equal  log- 

1  "Specimen  alterum  calculi  differentialis  in  dimetienda  Spirali  Logarithmica  Loxodromiis  Nautarum 

.  ,"  "per  J.B.,"  Acta  ernditorum,  1691,  pp.  282-283;  Opera,  tome  I,  Genevae,  1744,  pp.  442-443. 
Loria's  references  (/.  c,  p.  61)  to  Varignon  and  Bernoulli  are  distinctly  misleading.  In  1675  John  Collins 
used,  in  this  connection,  the  expression  "the  spiral  line  is  a  logarithmic  curve,"  Correspondence  0/  Scientific 
Men  of  the  Seventeenth  Century,  vol.  1,  1841,  p.  219;  [Quoted  in  full  in  a  later  footnote,  page  150]. 

In  more  than  one  place  Bernoulli  refers  to  the  logarithmic  spiral  as  the  'Spira  mirabilis,'  e.  g.  Opera,  tome 
1,  pp.  491,  497,  554;  also  Acta  eruditorum,  1692  and  1693. 

2  P.  Nicolas,  De  Novis  Spiralibus,  Exercitationes  Duae  .  .  In  posteriori  autem  agitur  de  alia  quadam 
spirali  a  prioribus  longe  diversa,  de  qua  V vallisius  &  V vrenius  insignes  Geometrae  scripserunt;  fc?  quae  illi  non 
altigere  circa  Tangentem  hujus  spiralis,  spatiorum  ilia  contentorum,  &  curvae  ipsius  dimensionem  absolvuntur. 
Tolosae,  1693.  "Exercitatio  II.  De  spiralibus  geometricis"  pp.  27-44.  Appendix,  pp.  45-51.  The  following 
quotation  from  page  27  may  be  given:  "Esto  curva  BCDEF  cujis  sit  talis  proprietas,  ut  omnes  radii  AB, 
AC,  AD,  AE,  AF  constituentes  angulos  aequales  in  centra  A  sint  inter  se  in  continua  proportione  Geometrica. 
Propter  hanc  insignen  proprietatam  curvam  BCDEF  vocamus  Spiralem  Geometricam  ut  distinguatur  a  Spirali 
communi  &  Archimedea,  cujus  proprietas  est  ut  radii  aequales  angulos  ad  centrum  sive  principium  Spiralis 
constituentes  sese  aequaliter  excedant,  ac  proinde  seryent  proportionem  Arithmeticam." 

3  E.  Halley,  Philosophical  Transactions,  1696.  The  lengths  of  segments  cut  off  from  a  radius  vector  between 
successive  whorls  of  the  spiral  form  a  geometric  progression. 

4  A  term  originating  with  R.  Cotes,  Philosophical  Transactions,  1714;  reprinted  after  the  death  of  Cotes 
in  his  Harmonia  Mensurarum,  Cantabrigiae,  1722  ("Aequiangula  spiralis,"  p.  19).  The  term  was  revived 
more  recently  by  Whitworth  in  Messenger  0/  Mathematics,  1862. 

6  See  two  letters  of  Collins,  one  undated  and  the  other  dated  Sept.  30,  1675,  in  Correspondence  of  Scientific 
Men  of  the  Seventeenth  Century  .  .  .  Vol.  1,  Oxford,  University  Press,  1841,  pp.  144,  218-19.  The  result  was 
first  given  in  print  by  E.  Halley,  in  Philosophical  Transactions,  1696. 

Cf.  F.  G.  M.,  Exercices  de  Geometrie  Descriptive,  4e  ed.,  Paris,  Mame,  1909,  pp.  824-6.  Chasles  showed 
(Apercu  historique,  etc.,  .  .  .  2e  ed.,  Paris,  1875,  p.  299)  that  if  the  logarithmic  curve  generates  a  surface  by 
revolving  about  its  asymptote,  and  if  this  asymptote  is  the  axis  of  a  helicoidal  surface,  the  two  surfaces  cut 
in  a  skew  curve  whose  orthogonal  projection  on  a  plane  perpendicular  to  the  asymptote  is  a  logarithmic 
spiral.  See  also  H.  Molins,  Memoires  de  I'academie  des  sciences  inscriptions  et  belles-lettres  de  Toulouse,  tome 
7  (sem.  2),  1885,  p.  293  f.;  tome  8,  1886,  pp.  426.  That  the  logarithmic  spiral  is  a  projection  of  a  certain 
"elliptic  logarithmic  spiral"  was  shown  in  W.  R.  Hamilton,  Elements  of  Quaternions,  London,  1866,  pp. 
382-3.  For  other  quaternion  discussion  of  the  logarithmic  spiral  see  H.  W.  L.  Hime,  The  Outlines  of  Qua- 
ternions, London,  1894,  pp.  171— 3. 

6  Cf.  Turquan,  "Demonstrations  elementaires  de  plusieurs  proprietes  de  la  spiral  logarithmique,"  Nouvelles 
annates  de  mathematiques,  tome  5,  1846,  pp.  88-97.  "Note"  by  Terquem  on  page  97. 

7  J.  Wallis,  Tractatus  Duo,  1659,  pp.  106-107;  also  Opera,  tome  1,  1695,  pp.  559-561- 

8  Paragraph  9  of  an  article  in  Acta  eruditorum,  May,  1692,  entitled  "Lineae  cycloidales,  evolutae,  ante- 
volutae,  causticae,  anti-causticae,  peri-causticae.  Earum  usus  et  simplex  relatio  ad  se  invicem.  Spira  mira- 
bilis. Aliaque  per  LB."  Cf.  Oeuvres  Completes  de  Christian  Huygens.  Tome  10.  La  Haye,  1905,  p.  119. 
The  center  of  curvature  at  a  point  on  a  logarithmic  spiral  is  the  extremity  of  the  polar  subnormal  of  the 
point. 


148 


DYNAMIC  SYMMETRY 


arithmic  spiral;1  (d)  the  caustics  by  reflection  and  refraction  of  a  logarithmic  spiral  for  rays 
emanating  from  the  pole  as  a  luminous  point  are  equal  logarithmic  spirals. 

The  discovery  of  such  "perpetual  renascence"  of  the  spiral  delighted  Bernoulli.  "Warmed 
with  the  enthusiasm  of  genius  he  desired,  in  imitation  of  Archimedes,  to  have  the  logarithmic 
spiral  engraved  on  his  tomb,  and  directed,  in  allusion  to  the  sublime  tenet  of  the  resurrection 
of  the  body,  this  emphatic  inscription  to  Tje  affixed — Eadem  mutata  resurgo."2  The  engraved 
spiral  (very  inaccurately  executed)  and  inscription,  in  accordance  with  Bernoulli's  desire,  may 
be  seen  to-day  on  his  tomb  in  the  cloister  of  the  cathedral  at  Basel.3 

The  logarithmic  spiral  appears  in  three  propositions  of  Newton's  "Principia"  (1687).4  From 
the  first  there  develops  that  if  the  force  of  gravity  had  been  inversely  as  the  cube,  instead  of 
the  square,  of  the  distance,  the  planets  would  have  all  shot  off  from  the  sun  in  "diffusive  log- 
arithmic spirals."5  In  the  second  proposition  Newton  showed  that  the  logarithmic  spiral  would 
also  be  described  by  a  particle  attracted  to  the  pole  by  a  force  proportional  to  the  square  of  the 
density  of  the  medium  in  which  it  moves,  while  this  density  is  at  each  point  inversely  propor- 
tional to  its  distance  from  the  pole.  In  the  third  proposition  the  second  was  generalized  by 
the  substitution  of  "inversely  proportional  to  any  power  of  its  distance"  for  "inversely  pro- 
portional to  its  distance" — a  result  which  has  been  attributed  to  Jacques  Bernoulli  (for  exam- 
ple, by  Gomes  Teixeira,  /.  c). 

There  is  also  considerable  discussion  of  the  logarithmic  spiral  by  Guido  Grandi  in  various 
parts  of  his  Geometria  Demonstratio  Theorematum  Hugenianorum  circa  Logisticam  seu  Log- 
arithmicam  Lineam  .  .  .  ,  Florentiae,  1701.6  A  section  in  the  first  chapter  deals  with 
"spiralio  logarithmicae  per  duos  motus  descriptio,"  and  points  are  found  (page  8)  "in  Spirali 
Logistica,  aliis  Spiralis  Logarithmicae,  quibusdam  Spiralis  Geometricae  nomine  appellata" 
(evidently  referring  to  P.  Nicolas,  /.  c).  In  a  letter  to  Ceva,  printed  at  the  end  of  the  vol- 
ume, the  gauche  spiral  cutting  the  generators  of  a  right  circular  cone  under  a  constant,  angle 
was  studied  for  the  first  time,  and  it  was  shown,  by  purely  geometric  methods,  that  this  spiral 
may  be  projected  into  a  logarithmic  spiral. 

In  a  memoir  read  by  Pierre  Varignon  before  the  French  Academy  in  17047  he  discussed  a 
transformation  equivalent  to  x  =  p,  y  =  I  to,  where  p  and  o>  are  the  polar  coordinates  of  the 
point  corresponding  to  (x,y),  and  /  is  a  constant.  Varignon  found,  in  particular,  that  from  the 

logarithmic  curve  x'h  =  ey  is  derived  the  logarithmic  spiral  p  =  e    h    .  So  also,  if  /  =  1,  the 

1  The  »th  positive  pedal  of  the  spiral  p  =  ke<8  with  respect  to  the  pole  is 

p  —  k  sin  <p.e    V2      ' .e 

(J.  Edwards,  Elementary  Treatise  on  the  Differential  Calculus,  3d  edition,  London,  Macmillan,  1896,  p.  167). 

2  Acta  eruditorum,  1706,  p.  44.  Cf.  Acta  eruditorum,  1692,  p.  212;  also  Opera,  tome  1,  Genevae,  1744,  p. 
502,  and  p.  30  of  "Vita." 

3  Cf.  L.  Isely,  "Epigraphies  tumulaires  de  mathematiciens,"  Bull,  de  la  societe  des  sciences  naturelles  de 
Neucbatel,  tome  27,  1899,  p.  171.  See  also  W.  W.  Rupert,  Famous  Geometrical  theorems  and  Problems  (Heath's 
Mathematical  Monographs,  part  4),  Boston,  1901,  p.  99. 

4  Book  I,  proposition  9,  and  book  II,  propositions  15  and  16. 

6  The  hodograph  of  an  equiangular  spiral  is  an  equiangular  spiral  (W.  Walton,  Collection  of  Problems  in 
Illustration  of  the  Principles  of  Theoretical  Mechanics,  3d  ed.,  Cambridge,  1876,  p.  296).  In  a  chapter  on  elec- 
tromagnetic observations  in  J.  C.  Maxwell's  Treatise  on  Electricity  and  Magnetism  (vol.  2,  Oxford,  Claren- 
don Press,  1873,  pp.  336-8)  the  discussion  calls  for  the  investigation  of  the  motion  of  a  body  subject  to  an 
attraction  varying  as  the  distance  and  to  a  resistance  varying  as  the  velocity.  This  leads  to  the  reproduction 
of  Tait's  application  (Proc.  Royal  Society  of  Edinburgh,  vol.  6,  1867,  p.  221  f.)  of  the  principle  of  the  hodo- 
graph to  investigate  this  kind  of  motion  by  means  of  the  logarithmic  spiral. 

"If  a  particle  be  describing  a  logarithmic  spiral  under  the  action  of  a  force  to  the  pole,  and  simultaneously 
the  law  of  force  be  altered  to  the  inverse  biquadrate  and  the  velocity  to  \/f  X  its  previous  value,  the  particle 
will  proceed  to  describe  a  cardioide."  Purkiss,  Messenger  of  Mathematics,  vol.  2,  1864.  For  other  results  of 
this  type,  involving  the  spiral,  see  Newton's  Principia,  first  book,  sections  I— III,  with  notes  and  illustra- 
tions by  P.  Frost,  London,  1880,  p.  203. 

6  Also  in  Christiani  Hugenii  Zuelechemi  .  .  .  Opera  Reliqua,  tome  1,  Amstelodami,  1728,  pp.  136-288. 

7  "Nouvelle  formation  de  spirales,"  Histoire  de  1' academic  royale  des  science,  annee  1704,  Paris,  1706,  pp. 
69-131;  see  especially  pp.  H3f. 


DYNAMIC  SYMMETRY 


149 


sine  curve  x  =  s\ny  becomes  a  circle.  In  recent  times  this  latter  transformation  has  been  em- 
ployed in  plotting  alternating  voltage  and  current  curves.1 

In  1892  I.  Stringham  showed2  that  if  the  logarithmic  spiral  is  properly  defined  as  a  geometric 
locus  it  may  be  used  for  defining  the  logarithm  and  demonstrating  its  properties,  which  lead  to 
a  classification  of  logarithmic  systems.  This  classification  was  somewhat  modified  by  M.  W. 
Haskell  and  I.  Stringham.3 

Cremona's  discussion  of  the  logarithmic  spiral,  and  how  it  may  serve,  when  drawn,  for  the 
solution  of  problems  involving  extraction  of  roots4  (higher  than  the  second)  should  not  be  for- 
gotten. Then  there  is  A.  Steinhauser's  Die  Elememe  des  graphischen  Rechnens  mit  besonderer 
Beriicksichtigung  der  logarithmischen  Spirale.  Eine  Einleitung  zur  Construction  algebraischer  und 
transcendenter  Ausdrucke  fur  Bau-  und  Maschinen-Techniker5 — Equiangular  spirals  appear 
as  "tie-lines"  and  "strutt-lines"  in  a  problem  of  W.  J.  Ibbetson's  Elementary  treatise  on  the 
Mathematical  Theory  of  Perfectly  Elastic  Solids* — There  is  also  the  little  known  but  notable 
paper,  published  by  James  Clerk  Maxwell  when  only  eighteen  years  of  age,7  which  contains 
several  properties  of  logarithmic  spirals.  Some  quotations  follow: 

"The  involute  of  the  curve  traced  by  the  pole  of  a  logarithmic  spiral  which  rolls  upon  any 
curve  is  the  curve  traced  by  the  pole  of  the  same  logarithmic  spiral  when  rolled  on  the  involute 
of  the  primary  curve."  (Page  524  [10].) 

"The  method  of  finding  the  curve  which  must  be  rolled  on  a  circle  to  trace  a  given  curve  is 
mentioned  here  because  it  generally  leads  to  a  double  result,  for  the  normal  to  the  traced  curve 
cuts  the  circle  in  two  points,  either  of  which  may  be  a  point  in  the  rolled  curve. 

"Thus,  if  the  traced  curve  be  the  involute  of  a  circle  concentric  with  the  given  circle,  the  rolled 
curve  is  one  of  two  similar  logarithmic  spirals."  (Page  529  [16].)  (Often  attributed  to  Haton  de 
la  Goupilliere.) 

"If  any  curve  be  rolled  on  itself,  and  the  operation  repeated  an  infinite  number  of  times,  the 
resulting  curve  is  the  logarithmic  spiral."  The  curve  which  being  "rolled  on  itself  traces  itself 
is  the  logarithmic  spiral,"  (Page  532  [19].) 

"When  a  logarithmic  spiral  rolls  on  a  straight  line  the  pole  traces  a  straight  line  which  cuts 
the  first  line  at  the  same  angle  as  the  spiral  cuts  the  radius  vector."  (Page  535  [23].)  (Often  at- 
tributed to  Catalan.) 

Among  many  other  results  the  following  may  be  noted:  Haton  de  la  Goupilliere  proved8  that 
the  logarithmic  spiral  is  the  only  curve  whose  pedal  with  respect  to  a  given  pole  is  an  equal 
curve  which  can  be  brought  into  coincidence  with  the  first  by  a  rotation  about  the  pole — Cesaro 


1  For  example:  D.  C.  Jackson  and  J.  P.  Jackson,  Alternating  Currents  and  Alternating  Current  Machinery , 
New  edition,  New  York,  1917,  pp.  13-15.  The  discussion  in  this  connection  seems  to  have  originated  with 
C.  P.  Steinmetz,  Trans.  Amer.  Inst.  Electrical  Engs.,  vol.  10,  p.  527;  Elektrotechnische  Zeitschrift,  June  20, 
1890. 

2  I.  Stringham,  "A  classification  of  logarithmic  systems,"  American  Journal  of  Mathematics,  vol.  14,  pp. 
187-194. 

3  Bulletin  of  the  New  York  Mathematical  Society,  vol.  2,  pp.  164-170, 1893.  See  also  I.  Stringham,  Uni  planar 
Algebra,  San  Francisco,  1893. 

4  L.  Cremona,  Graphical  Statics.  Translated  by  T.  H.  Beare,  Oxford,  Clarendon  Press,  1890,  pp.  59-64. 
Italian  edition,  Torino,  1 874,  pp.  39-42.  The  xylonite  logarithmic  spiral  curve  (eight  inches  in  width)  sold  by 
Keuffel  &  Esser  Co.,  New  York,  furnishes  the  means  for  accurately  and  rapidly  drawing  the  curve.  The 
curvature  gradually  changing  it  is  peculiarly  adapted  for  fitting  to  any  part  of  a  given  curve.  It  assists  in 
the  rapid  determination  of  the  center  of  curvature  of  a  given  part  of  the  curve,  and,  hence,  in  drawing  evo- 
lutes  and  equidistant  curves.  An  eight-page  pamphlet  by  W.  Cox  {The  logarithmic  spiral  curve  and  description 
of  its  uses,  1 891)  accompanies  the  instrument.  Eugene  Dietzgen  &  Co.,  Chicago,  manufactured  a  similar 
celluloid  instrument  and  a  ten-page  pamphlet  descriptive  of  its  use  was  written  by  E.  M.  Scofield,  and 
entitled  The  logarithmic  spiral  curve  (Chicago,  1892). 

6  Wien,  1885;  especially  pp.  40-75. 

6  London,  1887,  p.  322. 

7  "On  the  Theory  of  Rolling  Curves,"  Transactions  of  the  Royal  Society  of  Edinburgh,  vol.  16,  part  V, 
1849,  pp.  519-40.  [The  Scientific  Papers  of  J.  C.  Maxwell,  edited  by  W.  D.  Niven,  vol.  1,  Cambridge,  1890, 
pp.  4-29.]  Loria,  Gomes  Teixeira,  and  Wieleitner  seem  to  be  equally  ignorant  of  this  paper. 

8  journal  de  mathematique  pures  et  appliquees,  tome  11  (2),  1866,  pp.  329-336. 


i5o 


DYNAMIC  SYMMETRY 


discussed  the  tractrix  and  logarithmic  spiral  as  correlative  figures1 — From  logarithmic  spirals 
H.  Dittrich  derived2  (according  to  Loria,  /.  c.)  sum  and  difference  spirals  which  he  used  for 
geometrical  exposition  of  hyperbolic  functions — If  a  logarithmic  spiral  roll  on  a  straight  line 
the  locus  of  its  center  of  curvature  at  the  point  of  contact  is  another  straight  line  (A.  Mann- 
heim, 1859) — The  involutes  of  a  logarithmic  spiral  are  equal  spirals  (which  is  really  the  same  as 
Bernoulli's  result  for  evolutes)— The  inverse  of  a  logarithmic  spiral  with  respect  to  its  pole  is  an 
equal  spiral  with  the  same  pole — Coplanar  logarithmic  spirals  and  their  orthogonal  trajectories, 
which  are  again  coplanar  logarithmic  spirals,  come  up  (1)  in  the  discussion  of  loxodromic 
substitutions3  and  (2)  in  conformal  representations.4  As  a  consequence  of  a  general  theory  rela- 
tive to  linear  transformations  F.  Klein  and  S.  Lie  obtained  the  following  result:5  The  loga- 
rithmic spiral  is  its  own  polar  reciprocal  with  respect  to  any  equilateral  hyperbola  which  has  its 
center  at  the  pole  and  is  tangent  to  the  spiral. 

In  1833  T.  Olivier  described  to  the  Societe  Philomathique,  Paris,  "un  compass  simple  per- 
mettant  de  traces  toutes  les  spirales  logarithmiques,"6  and  i  n  a  letter  written  by  Collins  for  Tschirn- 
haus,  Sept.  30,  1675,7  reference  is  made  to  "an  instrument  invented  by  M.  Tschirnhaus"  and  its 
connection  with  the  logarithmic  spiral. 

The  most  practical  form  of  a  ship's  anchor  was  discussed  in  1796  by  F.  H.  Chapman,  vice- 
admiral  in  the  Swedish  Marine.8  He  found  that  the  best  form  for  each  of  the  barbed  arms  would 
be  an  arc  of  a  logarithmic  spiral  cutting  the  shank  of  the  anchor  at  an  angle  of  670  30'. 


1  Mathesis,  tome  2,  1882,  pp.  217-219. 

2  H.  Dittrich,  Die  logarithmische  Spirale,  Progr.  Breslau,  1872. 

3  F.  Klein  and  R.  Fricke,  V orlesungen  iiber  die  Theorie  der  elliptischen  Moduljunctionen,  Band  I,  Leipzig, 
Teubner,  1890,  p.  168. 

4  G.  Holzmiiller,  Einfiihrung  in  die  Theorie  der  isogonalen  Verwandtschaften  und  der  conformen  Abbildung, 
Leipzig,  Teubner,  1 882,  pp.  65,  238-241 ;  and  "Ueber  die  logarithmische  Abbildung  und  die  aus  ihr  entspring- 
enden  Curvensysteme,"  Zeitschrift  fiir  Mathematik  und  Physik,  Band  16,  1871,  pp.  269-289. 

5  Mathematische  Annalen,  Band  4,  1871,  p.  77.  Cf.  Encyklopddie  der  mathematischen  Wissenschaften,  Band 
IIL,  Leipzig,  1903,  pp.  210,  212;  also  Clebsch-Lindemann,  Vorlesungen  iiber  Geometrie,  Band  I,  Leipzig, 
Teubner,  1876,  p.  995. 

6  This  description  may  be  found  in  T.  Olivier,  Complements  de  geometrie  descriptive,  Paris,  1845,  P-  445- 
See  also  T.  Olivier,  Memoires  de  geometrie  descriptive,  Paris,  1851,  p.  284. 

7  This  letter  is  printed  in  Correspondence  of  Scientific  Men  of  the  Seventeenth  Century,  vol.  1,  Oxford,  1841. 
The  paragraphs  of  special  interest  in  this  connection  are  as  follows:  "As  to  the  instrument  invented  by  M. 
Tschirnhaus  for  dividing  an  angle  in  ratione  data,  we  suppose  he  gives  an  angle  as  geometers  do,  ready 
drawn  by  accident  or  at  pleasure,  and  then  I  conceive  it  an  instrument  worthy  the  author:  whereas  here 
(so  far  as  I  know)  we  have  nothing  but  the  old  mechanism,  viz.  to  measure  the  angle  in  degrees  first,  by  aid 
of  a  sector  or  opening  joint,  and  then  set  off  the  part  proportional  by  aid  of  an  arch  or  line  of  chords,  which 
one  of  the  legs  may  draw  after  it,  which  part  proportional  may  be  attained  by  a  sliding  scale  with  logca' 
lines  upon  it,  which  may  be  annexed  to  the  other  leg;  but  here  I  will  a  little  enlarge  on  the  use  of  M.  Tschirn- 
haus's  invention. 

"We  have  an  instrument  called  the  serpentine  line,  or,  as  Oughtred  terms  it  the  circles  of  proportion,  in 
the  use  whereof,  in  relation  to  compound  interest,  it  is  often  required  to  divide  an  angle  in  ratione  data,  or  an 
angle  being  given  to  enlarge  it  in  ratione  data.  Moreover,  conceive  the  eye  at  the  south  pole,  projecting  the 
loxodromia  or  rumb  of  a  ship's  course  on  the  earth,  on  a  plane  touching  the  sphere  at  the  north  pole,  the 
projected  curve  will  be  a  spiral  line,  in  which,  if  the  polar  rays  PE,  PD,  PC,  P/E,  [the  figure  of  the  letter  is 
omitted]  make  equal  angles  at  the  pole  P,  those  rays  will  be  in  continual  geometrical  proportion;  and  con- 
ceiving a  circle  described  upon  P  as  a  centre,  the  equal  segments  of  the  arch  in  the  circumference,  made  by 
the  polar  rays,  will  be  an  arithmetical  progression,  suited  to  a  geometrical  one;  consequently  the  spiral 
line  is  a  logarithmic  curve  and  from  hence  the  meridian  line  of  the  true  sea  chart  may  be  demonstrated  to 
be  a  line  of  logarithmic  tangents,  and  the  spiral  line  with  M.  Tschirnhaus's  angular  instrument,  makes  the 
mesolabe  [an  instrument  for  finding  mean  proportionals  between  two  numbers],  which  our  late  learned 
Oughtred  said  was  hitherto  tenebris  obvolutum. 

"To  rectify  or  straighten  this  spiral,  or  part  of  it,  as  EJE,  is  all  one  effect  as  to  draw  a  touch-line  to  it, 
or  to  find  the  rumb  between  two  places  whose  latitudes  and  difference  of  longitude  are  given  which  to  per- 
form in  lines  is  a  proposition  of  great  use,  and  hitherto  wanting  in  navigation,  and  depends  on  the  quadrature 
of  the  hyperbola,  as  Dr.  Barrow,  at  my  instance,  proved  in  his  Geometrical  Lectures.  Moreover  such  a 
spiral,  being  once  well  described,  may  serve  to  take  away  the  use  of  compasses  in  Galileus  or  our  Gunter's 
sector  or  joint  for  proportions,  all  which  I  thought  not  impertinent  to  hint." 

8  "Om  ratta  Formen  pa  Skepps-Ankrar,"  Svensk.  Vetensk.  Academ.  nya  Hand/.,  1796,  Vol.  17,  pp.  1-24. 


DYNAMIC  SYMMETRY 


The  distinctive  properties  of  the  logarithmic  spiral  which  permit  it  to  be  used  for  lines  of 
pitch  of  cams  and  non-circular  wheels1  are:  (a)  that  the  difference  of  radii  vectores  of  the  ends 
of  equal  arcs  is  constant;  (b)  the  curve  cuts  radii  vectores  under  a  constant  angle.  For  these 
reasons  two  equal  logarithmic  spirals  may  roll  together  with  fixed  poles  and  a  fixed  distance 
between  the  poles.  Two  arcs  (not  necessarily  equal)  of  logarithmic  are  required  for  the  complete 
line  of  pitch  of  a  wheel,  but  any  even  number  of  arcs  may  be  used.  A  wheel  with  three  lobes  may 
act  on  a  wheel  with  two,  which  in  turn  may  act  on  a  unilobe  wheel.  Even  with  two  reacting 
wheels  with  the  same  number  of  lobes  there  are  varying  velocity  ratios  having  maximum  and 
minimum  values  for  the  rates  of  rotation  of  the  shafts. 

The  first  definite  suggestion  connecting  the  logarithmic  spiral  with  organic  spirals  seems  to 
have  been  made  by  Sir  John  Leslie  in  his  Geometrical  Analysis  and  Geometry  of  Curve  Lines? 
After  proving  that  the  involutes  of  a  logarithmic  spiral  are  logarithmic  spirals  he  remarks: 
"The  figure  thus  produced  by  a  succession  of  coalescent  arcs  described  from  a  series  of  interior 
centers  exactly  resembles  the  general  form  and  the  elegant  septa  of  the  Nautilus."*  The  aptness 
of  this  remark  has  been  long  since  established.  One  of  the  earliest  mathematical  discussions  of 
organic  logarithmic  spirals  was  by  Canon  Moseley,  "On  the  Geometrical  Forms  of  Turbinated  and 
Discoid  Shells"4 — a  paper  written  more  than  eightyyears  ago  which  is  one  of  the  classics  of  natural 
history.  In  "turbinate"  shells  we  are  no  longer  dealing  with  a  plane  spiral  as  in  the  nautilus  but  with 
a  gauche  spiral  on  a  right  circular  cone  cutting  the  generators  at  a  constant  angle  and  such  that 
along  a  generator  the  line-segments  between  successive  whorls  form  a  geometric  progression.5 
For  mathematical  and  other  details  of  Moseley's  work  as  well  as  of  that  of  many  others,  on 
univalve  and  bivalve  shells,  Thompson's  book,  with  its  many  exact  references  to  the  literature 
of  the  subject,  should  be  consulted.  One  notable  work  which  Thompson  appears  to  have  over- 
looked is  Haton  de  la  Goupilliere,  "Surfaces  Nautiloides."6 

In  the  field  of  leaf  arrangement  or  phyllotaxis  discussion  of  the  theories  of  A.  H.  Church7 
and  Cook  evolved  from  observations  of  arrangements  in  logarithmic  spirals  of  florets  of  sun- 
flowers, pine  cones,  and  other  growths,  should  be  read  in  connection  with  Thompson's  criticisms. 
The  fine  sunflower  photograph  by  H.  Brocard8  ought  to  be  compared  with  those  by  Church. 

Abridged  and  translated  in  Annalen  der  Physik  (Gilbert),  Band  6,  Halle,  1800:  "Von  der  richtigen  Form  der 
Schiffsanker,"  pp.  81-95. 

1  W.  J.  M.  Rankine,  Manual  of  Machinery  and  Millwork,  London,  1869,  pp.  99-102; 
C.  W.  MacCord,  Kinematics,  New  York,  1883,  pp.  47-50; 

F.  Reuleaux,  Lehrbuch  der  Kinematik,  Band  2:  Die  praktischen  Beziehungen  der  Kinematik  zu  Geometrie 
und  Mechanik,  Braunschweig,  1900,  pp.  473,  542-544; 

P.  Schwamb  and  A.  L.  Merrill,  Elements  of  Mechanism,  New  York,  1 913,  pp.  32-36; 
R.  F.  McKay,  The  Theory  of  Machines,  London,  1915,  pp.  218-222. 

F.  DeR.  Furman,  "Cam  design  and  construction,"  American  Machinist,  vol.  51,  pp.  695-698,  Oct.  9,  1919. 

2  Edinburgh,  1821,  p.  438. 

3  For  pictures  of  the  nautilus  pompilius  see  pp.  494,  581,  582  of  D.  W.  Thompson,  On  Growth  and  Form, 
Cambridge  University  Press,  1917,  and  also  pp.  57,  457  of  T.  A.  Cook,  The  Curves  of  Life,  London,  Constable, 
1914.  This  latter  work  contains  many  beautiful  illustrations  and  logarithmic  spiral  forms  are  specially  dis- 
cussed on  pages  60-63,  4I3_42Ii  another  work  by  the  same  author,  Spirals  in  Nature  and  Art,  London, 
Murray,  1903,  has  some  good  illustrations. 

4  Philosophical  Transactions  of  the  Royal  Society,  London,  Vol.  128,  1838,  pp.  351-370. 

5  As  early  as  1701  Guido  Grandi  showed,  /.  c,  as  already  noted,  that  the  orthogonal  projection  ot  this 
spiral  on  a  plane  perpendicular  to  the  axis  of  the  cone  is  a  logarithmic  spiral.  The  gauche  spiral  has  been 
studied  by  Th.  Olivier  (who  called  it  the  conical  logarithmic  spiral),  Developpements  de  geometrie  descrip- 
tive, 1843,  pp.  56-76;  by  P.  Serret,  Theorie  nouvelle  geometrique  et  mecanique  des  lignes  a  double  courbure, 
i860,  p.  1 01;  etc.  A  number  of  results  are  collected  by  Gomes  Teixiera,  /.  c,  pp.  396-400.  _ 

For  other  surfaces  involving  the  logarithmic  spirals  reference  should  be  given  to  the  very  interesting  pages 
232_3!3  °f  G.  Holzmiiller,  Elemente  der  Stereometrie,  Drifter  Teil,  Leipzig,  Goschen,  1902,  on  logarithmic 
spiral  tubular  surfaces  and  their  inverses. 

6  This  occupies  almost  the  whole  of  the  third  volume  of  Annaes  scientifcos  da  academia  polytechnica  do 
Porto,  Coimbra,  1908.  Cf.  L' Intermediate  des  mathematiciens,  1900,  tome  7,  p.  40;  1901,  tome  8,  pp.  167, 
314;  1910,  tome  17,  p.  155. 

7  A.  H.  Church,  On  the  Relation  of  Phyllotaxis  to  Mechanical  Law,  London,  Williams  and  Norgate,  1904. 

8  In  L' Intermediate  des  mathematiciens,  1909,  and  in  H.  A.  Naber,  Das  Theorem  des  Pythagoras,  Haarlem, 
Visser,  1908,  opposite  p.  80. 


152 


DYNAMIC  SYMMETRY 


II.   Golden  Section. 

In  the  "Elements"  of  Euclid  (who  flourished  about  300  B.  C),  the  following  propositions 
occur:  (1)  "To  cut  a  given  straight  line  so  that  the  rectangle  contained  by  the  whole  and  one 
of  the  segments  is  equal  to  the  square  on  the  remaining  segment"  (Book  II,  proposition  11); 
(2)  "To  cut  a  given  finite  line  in  extreme  and  mean  ratio"  (Book  VI,  proposition  30).1  While 
these  propositions  are  equivalent  in  statement  the  methods  of  construction  given  by  Euclid  are 
quite  different.  There  can  be  little  doubt  that  the  construction  in  the  second  is  due  to  Euclid 
and  in  the  first  to  the  Pythagoreans  (fifth  century  B.  C).  The  result  is  used  "To  construct  an 
isosceles  triangle  having  each  of  the  angles  at  the  base  double  of  the  remaining  one"  (Elements, 
Book  IV,  10)  and  this  leads  to  the  construction  of  a  regular  pentagon  (Book  IV,  11). 

In  the  Elements,  book  XIII,  the  first  five  propositions,  which  are  preliminary  to  the  con- 
struction and  comparison  of  the  five  regular  solids,  and  deal  with  properties  of  a  line  segment 
divided  in  extreme  and  mean  ratio,  are  usually  attributed  to  Eudoxus,  who  flourished  about 
365  B.  C.  Proclus  tells  us  that  Eudoxus  "greatly  added  to  the  number  of  the  theorems  which 
Plato  originated  regarding  the  section";  scholars  agree  that  "the  section"  refers  to  the  division 
in  extreme  and  mean  ratio. 

The  so-called  book  XIV  of  Euclid's  Elements,  written  by  Hypsicles  of  Alexandria  between 
200  and  100  B.  C,  contains  some  results  concerning  "the  section." 

In  recent  times  the  name  golden  section  has  been  applied  to  the  division  of  a  line  segment 
as  above2  in  the  ratio  (V5  —  1)  :  2.  Terquem  believed  that  the  expression  "extreme  and  mean 
ratio"  (which  is  an  exact  translation  of  Euclid's  Greek  phrase)  is  "une  reunion  de  mots  ne  pre- 
sentant  aucun  sens,"3  and  following  J.  F.  Lorenz  (1781)  employed  the  term  "continued  section." 
Terquem  has  also  suggested:4  "diviser  une  droite  decagonalement."  Leslie  introduced  the  term 
"medial  section."5  "Divine  proportion"  was  used  by  Fra  Luca  Pacioli  in  15096  and  possibly 
earlier  by  Pier  della  Francesca;7  "sectio  divina"  and  "proportio  divina"  occur  in  the  writings 
of  Kepler. 

1  These  enunciations  are  taken  from  The  Thirteen  Books  of  Euclid's  Elements  translated  with  introduction 
and  commentary  by  T.  L.  Heath,  3  vols.,  Cambridge,  at  the  University  Press,  1908.  For  statements  in  con- 
nection with  our  discussion  see  particularly,  Vol.  1,  pp.  137,  403;  Vol.  2,  p.  99;  Vol.  3,  p.  441. 

2  The  earliest  instances  which  I  find  of  the  use  of  the  term  golden  section  are  in  J.  Helmes,  "Eine  einfachere, 
auf  einer  neuen  Analyse  beruhende  Auflosung  der  sectio  aurea,  nebst  einer  kritischen  Beleuchtung  der 
gewohnlichen  Auflosung  und  der  Betrachtung  ihres  padagogischen  Werthes."  Archiv  der  Mathematik,  Gru- 
nert,  Band  4,  1844,  pp.  15-22;  in  A.AViegand,  Geometrische  Lehrsdtze  und  Aufgaben,  Band  2,  1.  Abtheilung, 
Halle,  1847,  p.  142;  and  also  in  A.  Wiegand,  Der  allgemeine  goldene  Schnitt  und  sein  Zusammenhang  mit  der 
harmonischen  Theilung.  .  .  Halle,  1849. 

Much  negative  evidence  seems  to  indicate  that  the  term  'golden  section'  was  originated  within  the  thirty 
years  1815-1844.  For  example,  it  is  not  mentioned  in  Kliigel-Mollweide's  Mathematisches  Wbrterbuch, 
which  contains  so  many  references  to  the  literature  of  different  topics.  We  do,  however,  find  the  following 
(Erste  Abtheilung,  vierter  Theil,  Leipzig,  1823,  p.  363):  "Die  Aufgabe  bey  Eukleides  II,  11,  oder  VI.  30, 
ist  sonst  auch  bisweilen  sectio  divina  genannt." 

3  Nouvelles  annates  de  mathematiques,  Paris,  tome  12,  1853,  p.  38. 

4  Journal  de  mathematiques  pures  et  appliquees,  Paris,  tome  3,  1838,  p.  98. 

6  J.  Leslie,  Elements  of  Geometry,  geometrical  Analysis  and  plane  'Trigonometry,  Edinburgh,  1809,  p.  66. 

6  Divina  Proportione  opera  a  tutti  gli  ingegni  perspicaci  e  curiosi  necessaria  que  ciaseum  studioso  di  philoso- 
phia:  prospettiva,  pictura,  sculptura,  architectura:  musica:  e  altre  matematice  .  .  .  V enetiis  .  .  I5°9-  Although 
not  printed  till  1509  the  manuscript  of  this  work  was  completed  in  1497.  The  geometrical  drawings  were 
made  by  Leonardo  da  Vinci;  cf.  G.  Libri,  Histoire  des  Sciences  math,  en  Italie,  tome  3,  Paris,  1840,  p.  144, 
note  2.  Another  edition  of  the  Latin  text  "herausgegeben,  iibersetzt  und  erlautert  von  C.  Winterberg" 
appeared  at  Vienna  (Graser)  1889.  Another  edition  1896,  6  +  367  pp.  A  full  analysis  of  Pacioli's  work  is  to 
be  found  in  A.  G.  Kastner,  Geschichte  der  Mathematik  .  .  .  Band  I,  Gottingern,  1796,  pp.  417-449.  See  also 
M.  Cantor,  Vorlesungen  iiber  Geschichte  der  Mathematik,  Band  2,  2.  Auflage,  Leipzig,  1900,  pp.  341  ff.,  347. 

7  It  has  been  shown  by  G.  Mancini  that  parts  of  Pacioli's  Divina  Proportione  were  taken  from  a  Vatican 
manuscript  by  Pier  della  Francesca.  See  (1)  G.  Pittarelli,  Atti  del  IV.  congresso  del  matematici,  tomo  3, 
Roma,  1909;  (2)  G.  Mancini,  "L'opera  'De  Corporibus  Regularibus'  di  Pietro  Franceschi  detto  Francesca 
usurpata  da  Fra  Luca  Pacioli"  (con  dodici  tavole)  Reale  accademia  dei  Lincei,  191 5.  See  review  by  F.  Cajori 
in  the  American  Mathematical  Monthly ,  Vol.  23,  1916,  p.  384.  (3)  G.  B.  de  Toni,  "Intorno  al  codice  sforzesco 
'De  divina  proportione*  di  Luca  Pacioli  e  i  disegni  geometrici  di  qust'  opera  attributi  a  Leonardo  da  Vinci, 
Modena  soc.  dei  naturalistic  e  matematici,  atti,  134,  191 1,  pp.  52-79. 


DYNAMIC  SYMMETRY 


153 


Pacioli's  work  was  doubtless  influential  in  inspiring  a  certain  amount  of  mysticism  in  the 
consideration  of  golden  section  by  later  writers.  In  a  work  published  in  1 569,  P.  Ramus  asso- 
ciates the  Trinity  with  the  three  parts  of  golden  section.  A  little  later  Clavius  wrote  of  its  "god- 
like proportions."  As  noted  above  Kepler  declared  himself  similarly.  He  said  also:  "Geometry 
has  two  great  treasures,  one  is  the  Theorem  of  Pythagoras,  the  other  the  division  of  a  line  into 
extreme  and  mean  ratio;  the  first  we  may  compare  to  a  measure  of  gold,  the  second  we  may 
name  a  precious  jewel."1 

In  the  Thirteenth  Century  Campanus  proved  (in  his  edition  of  Euclid's  Elements,  bk.  IX, 
prop.  16)  that  golden  section  was  irrational.  His  argument  (by  mathematical  induction)  was 
reproduced  in  algebraic  notation  by  Genocchi  and  by  Cantor.2 

There  is  an  interesting  passage  on  golden  section  by  Albert  Girard  in  his  edition  of  Stevin's 
works.3  Girard  gives  a  method  of  expressing  the  ratio  of  the  segments  of  a  line  (cut  in  golden 
section)  in  rational  numbers  that  converge  to  the  true  ratio.  For  this  purpose  he  takes  the 
sequence 

00  0,1,1,2,3,5,8,13,21,..'., 

every  term  of  which  (after  the  second)  is  equal  to  the  sum  of  the  two  terms  that  precede  it, 
and  says,  after  Kepler,  any  number  in  this  progression  has  to  the  following  the  same  ratios 
(nearly)  that  any  other  has  to  that  which  follows  it.  Thus  5  has  to  8  nearly  the  same  ratio  that 
8  has  to  13;  consequently  any  three  consecutive  numbers  such  as  8,  13,  21  nearly  express  the 
segments  of  a  line  cut  in  golden  section.  Since  the  fractions 

(1)  i  4  i  4  i'  _%r  ■  A3-  '  • 

1>  Y>  ^>         8>   l5>  2T> 

are  the  various  convergents  of  the  continued  fraction 

V5  -  I  1 


2  1 

1  +  ■ 

1 

1  +  

1 


Maupin  reasons  with  force  (after  taking  into  account  all  which  follows  in  the  note)  that  Girard 
was  probably  familiar  with  the  elements  of  continued  fractions.  Simson  interprets  Girard's 
reasoning  differently. 

For  mathematical  treatment  of  problems  in  golden  section,  in  ordinary  or  generalized  form, 
see  also  the  papers  by  C.  Thiry4  and  R.  E.  Anderson,5  E.  Catalan's  "Theoremes  et  Problemes  de 

1  Exact  references  to  sources,  and  some  quotations  from  originals,  are  given  in  (1)  J.  Tropfke,  Geschichte 
der  Elementar-Mathematik,  Band  2,  Leipzig,  Veit,  1903;  (2)  F.  Sonnenburg,  Der  goldne  Schnitt.  Beitrag 
zur  Geschichte  der  Mathematik  and  ihre  Anwendung.  (Progr.),  Bonn,  1 881.  (Not  always  reliable.)  Cf.  ftn. 
4»  P-  J55- 

2  Annali  di  scienze  matematiche  e  fisiche  (Tortolini),  vol.  6,  1855,  pp.  307-308;  also  M.  Cantor,  Vorle- 
sungen  ilber  Geschichte  der  Mathematik,  vol.  2,  2.  ed.,  1900,  pp.  105-106;  see  also  American  Mathematical 
Monthly,  vol.  25,  191 8,  p.  197,  and  Bulletin  of  the  American  Mathematical  Society,  vol.  15,  1909,  p.  408. 

3  Les  oeuvres  mathematiques  de  Simon  Stevin  .  .  .  le  tout  revu,  corrige  et  augmente  par  A.  Girard.  Leyde, 
1 634,  pp.  169-170.  The  passage  in  question  is  reprinted  with  commentary  in  G.  Maupin,  Opinions  et  Curiostes 
touchantla  Mathematique  (deuxieme  serie),  Paris,  1902,  pp.  203-209.  It  has  been  discussed  also  by  R.  Simson, 
Philosophical  Transactions,  1753,  vol.  48,  pp.  368-377;  see  "Reflexions  sur  la  preface  d'un  memoire  de 
Lagrange  intitule:  'Solution  d'un  probleme  d'arithmetique'  "  by  J.  Plana,  Memoire  del/a  r.  accademia  d.  sci- 
enze di  Torino,  series  2,  vol.  20,  Torino,  1863,  especially  pp.  89-92. 

4  C.  Thiry,  "Quelques  proprietes  d'une  droite  partagee  en  moyenne  et  extreme  raison,"  Mathesis,  1894, 
vol.  14,  pp.  22-24. 

6  "Extension  of  the  medial  section  problem  and  derivation  of  a  hyperbolic  graph,"  Proceedings  of  the 
Edinburgh  Mathematical  Society,  1897,  Vol.  15,  pp.  65-69. 


154 


DYNAMIC  SYMMETRY 


geometric  elementaire"1  and  Emsmann's  program2  containing  more  than  350  relations  and  prob- 
lems. 

In  the  nineteenth  century  the  literature  of  golden  section  is  by  no  means  inconsiderable.  It 
includes  at  least  a  score  of  separate  pamphlets  and  books  and  many  times  that  number  of  papers. 
In  numerous,  voluminous  and  rather  unscientific  writings  A.  Zeising3  finds  golden  section  the 
key  to  all  morphology  and  contends,  among  other  things,  that  it  dominates  both  archi- 
tecture and  music.  A  distinctly  new  line  was  set  under  way  by  Fechner  who  applied  scientific 
experimental  methods  to  the  study  of  aesthetic  objects.4  He  was  led  to  the  conclusion  that  the 
rectangle  of  most  pleasing  proportions  was  one  in  which  the  adjacent  sides  are  in  the  ratio  of 
parts  of  a  line  segment  divided  in  golden  section.6  There  are  some  paragraphs  on  "Golden  Sec- 
tion," by  J.  S.  Ames  in  Dictionary  of  Philosophy  and  Psychology*  edited  by  J.  M.  Baldwin.  In 
his  article  on  "The  aesthetics  of  unequal  division"7  P.  A.  Angier  discusses  earlier  contributions 
to  the  aesthetics  of  golden  section,  including  those  by  L.  Witmer8  (the  chief  investigator  in  the 
aesthetics  of  simple  forms  after  Fechner),  W.  Wundt,9  and  O.  Kulpe.10  The  subject  has  been 
treated  still  more  recently  by  M.  Dessoir11  and  J.  Volkelt.12 

Sir  Theodore  Cook  discusses13  golden  section  from  some  new  points  of  view  in  connection 
with  art  and  anatomy,  and  the  writings  of  F.  X.  Pfeifer14  remind  one  both  in  subject  matter  and 
style  of  treatment  of  Zeising's  publications. 

Neikes  defined  the  term  golden  section  for  different  units  (areas,  volumes — not  alone  line- 
segments)  such  that  the  smaller  part  is  to  the  larger  as  the  larger  is  to  the  whole.  With  Piazzi 
Smyth's  work  as  a  basis  he  applied  golden  section  to  an  unscientific  study  of  the  architecture  of 
the  Cheops  pyramid.15 

1  6e  ed.,  Paris,  1879,  pp.  261-263.  Some  of  these  properties  are  given  in  the  first  edition  of  this  work, 
which  was  really  written  by  H.  C.  de  La  Fremoire,  Paris,  1844. 

2  D.  H.  Emsmann,  Zur  sectio  aurea.  Materialien  zu  elementaren  namentlich  durch  die  Sectio  aurea  loslichen 
Constructions-aufgaben  etc.,  Progr.  Stettin,  1874  (Cf.  Zeitschrift  f.  math,  und  naturw.  Unterricht,  vol.  5, 
pp.  289-291). 

3  For  example  (1)  Neue  Lehre  von  den  Proportionen  des  menschlichen  Korpers  aus  einem  bisher  unerkannt 
gebliebenen,  die  game  Natur  and  Kunst  durchdringenden  morphologischen  Grundgesetze  entwickelt,  Leipzig, 
1854,  457  pp.;  particularly  pages  133-174;  (2)  Aesthetische  Forschungen,  Frankfort,  1855,  pp.  1791".  (3)  Das 
Normalverhaltnis  der  chemischen  und  morphologischen  Proportionen,  Leipzig,  1856,  114  pp.  and  the  post- 
humous work:  (4)  Der goldene  Schnitt,  Leipzig,  1884,  28  pp.  Cf.  S.  Giinther,  "Adolph  Zeising  als  Mathematik- 
er,"  Zeitschrift  fiir  Mathematik  und  Physik,  Historisch-literarische  Abtheilung,  Band  21,  1876,  pp.  157-165. 

4  G.  T.  Fechner,  Zur  experimentalen  Aesthetik,  Leipzig,  1 871;  also  Vorschule  der  Aesthetik,  Leipzig,  1876, 
pp.  i85f. 

5  C.  L.  A.  Kunze  speaks  of  "Rechteck  der  schonsten  Form"  in  his  Lehrbuch  der  Planimetrie,  Weimar, 
l%39>  p.  124.  A  reference  may  be  given  to  a  recent  discussion  of  "printer's  oblong"  and  "golden  oblong" 
in  H.  L.  Koopman,  "Printing  page  problems  with  geometric  solutions,"  The  Printing  Art,  Cambridge, 
Mass.,  191 1,  vol.  16,  pp.  353-356. 

6  New  York,  vol.  1,  1901,  p.  416. 

7  Harvard  Psychological  Studies,  vol.  1,  1903,  pp.  541-561. 

8  L.  Witmer,  "Zur  experimental  Aesthetik  einfacher  raumlicher  Formverhaltnisse"  Philosophische  Studi- 
en,  Leipzig,  vol.  9,  1893,  pp.  96-144,  209-263. 

9  W.  Wundt,  Grundziige  der  physiologischen  Psychologie,  Band  2,  4.  Auflage,  1893,  pp.  24of.  (See  also  Band 
3,  6.  Auflage,  191 1,  pp.  136L). 

10  O.  Kiilpe,  Outlines  of  Psychology,  translated  into  English  by  E.  P.  Titchener,  London,  1895,  pp.  253-255. 

11  M.  Dessoir,  Aesthetik  und  allgemeine  Kunstuissenschaft  in  den  Grundziigen  dargestellt,  Stuttgart,  1906, 
pp.  I24f,  176-177. 

12  J.  Volkelt,  System  der  Aesthetik,  Band  2,  Munchen,  1910,  pp.  33L 

13  T.  A.  Cook,  The  Curves  of  Life,  London,  Constable,  1914. 

14  (a)  "Die  Proportion  des  goldenen  Schnittes  an  den  Blattern  und  Stengeln  der  Pflanzen,"  Zeitschrift 
fiir  mathematischen  und  naturwissenschaftlichen  Unterricht,  1885,  vol.  15,  pp.  325-338;  (b)  Der  goldene 
Schnitt  und  dessen  Erscheinungsformen  in  Mathematik  Natur  und  Kunst,  Augsburg,  [1885],  3  +  232  pp. 
+  13  plates.  A  resume  of  this  work  given  by  O.  Willman  in  Lehrproben  und  Lehrgdnge  aus  der  Praxis  der 
Gymnasien  und  Realschulen,  1892  was  the  basis  of  E.  C.  Ackermann,  "The  Golden  Section,"  American 
Mathematical  Monthly,  1895,  vol.  2,  pp.  260-264.  Cf.  Zeitschrift  j.  math,  und  naturwiss.  Unterricht,  1887, 
vol.  18,  pp.  44-47,  605-612. 

15  H.  Neikes,  Der  goldene  Schnitt  und  ihre  Geheimnisse  der  Cheops  Pyramide,  Coin,  1907;  (reviewed  in  Jahr- 
buch  iiber  die  Fortschritte  der  Mathematik,  1907,  p.  526).  Pages  3-10:  "der  goldene  Schnitt";  pages  11-20: 
"die  Geheimnisse  der  Cheops  Pyramide."  C.  Piazzi  Smyth,  Life  and  Work  at  the  great  Pyramids,  1867. 


DYNAMIC  SYMMETRY  155 

III.    The  Fibonacci  Series. 

Foremost  among  mathematicians  of  his  time  was  Leonardo  Pisano  (also  known  as  Fibonacci), 
who  flourished  in  the  early  part  of  the  thirteenth  century.  His  greatest  work  is  Liber  abbaci 
"a  Leonardo  filio  Bonacci  compositus,  anno  1202  et  correctus  ab  eodem  anno  1228."  It  was 
first  printed  in  1857.1 

Among  miscellaneous  arithmetical  problems  of  the  twelfth  section  is  one  entitled  "How 
many  pairs  of  rabbits  can  be  produced  from  a  single  pair  in  a  year."2  It  is  supposed  (1)  that 
every  month  each  pair  begets  a  new  pair  which,  from  the  second  month  on,  becomes  productive; 
and  (2)  that  deaths  do  not  occur.  From  these  data  it  is  found  that  the  number  of  pairs  in  suc- 
cessive months  would  be  as  follows: 

(3)  h  2>  3,  S,  8>  l3,  2I>  34,  55>  89>  144.  233>  377- 

These  numbers  follow  the  law  that  every  term  after  the  second  is  equal  to  the  sum  of  the  two 
preceding  and  form,  according  to  Cantor,  the  first  known  recurring  series  in  a  mathematical 
work.  The  doubtful  accuracy  of  this  latter  statement  has  been  pointed  out  by  Giinther.3 

The  series  (3)  was  well  known  to  Kepler,  who  discusses  and  connects  it  with  golden  section 
and  growth,  in  a  passage  of  his  "De  nive  sexangula,"  161 1. 4  Commentaries  of  Girard  and  Simson, 
and  the  relation  of  the  series  to  a  certain  continued  fraction,  have  been  noted  above.  But  the 
literature  of  the  subject  is  very  extensive  and  reaches  out  in  a  number  of  directions.  In  what 
follows  un  will  be  regarded  as  the  (n  4-  i)st  term  of  what  we  shall  call  the  Fibonacci  series  (1); 
so  that  Uo  =  o,  ti\  =  «2  =  I,  #3  =  2,  .  .  .  For  reasons  which  shall  appear  later  the  names 
Lame  series,  and  Braun  or  Schimper-Braun  series,  have  been  also  employed  in  this  connection. 
Girard  observed,  /.  c,  that  the  three  numbers  un,  un+\,  Mn+^may  be  regarded  as  corresponding  to 
lengths  which  form  an  isosceles  triangle  of  which  the  angle  at  the  vertex  is  very  nearly  equal 
to  the  angle  at  the  center  of  the  regular  pentagon. 

The  relation  #n_i«n+i  —  uj  =  (  —  1)"  was  stated  in  1753  by  Simson  (/.  c).  It  was  to  this 
relation,  and  hence  to  the  Fibonacci  series  that  Schlegel6  was  led  when  he  sought  to  generalize 
the  well-known  geometrical  paradox  of  dividing  a  square  8x8  into  four  parts  which  fitted  to- 
gether form  a  rectangle  5  X  13.7  Catalan  found  (1879)  the  more  general  relation8  Un+i-pUn+\+P  — 
«„+i2  =  (—  i)"_p(kp)2,  from  which  may  be  derived  «„+i2  4-  un2  =  «2n+i  first  given,  along  with 


1  II  liber  Abbaci  di  Leonardo  Pisano  pubblicato  da  Baldassare  Boncompagni,  Roma,  MDCCCLVII. 
For  an  analysis  of  this  work  see  M.  Cantor,  V orlesungen  iiber  Geschichte  der  Mathematik,  Band  II,  3.  Auflage, 
Leipzig,  Teubner,  1900,  pp.  5-35. 

2  Pages  283-284. 

3  S.  Giinther,  Geschichte  der  Mathematik,  1.  Teil,  Leipzig,  Goschen,  1908,  p.  137. 

4  J.  Kepler,  Opera,  ed.  Frisch,  tome  7,  pp.  722-3.  After  discussions  of  the  form  of  the  bees'  cells  and  of  the 
rhombo-dodecahedral  form  of  the  seeds  of  the  pomegranite  (caused  by  equalizing  pressure)  he  turns  to  the 
structure  of  flowers  whose  peculiarities,  especially  in  connection  with  quincuncial  arrangement  he  looks 
upon  as  an  emanation  of  sense  of  form,  and  feeling  for  beauty,  from  the  soul  of  the  plant.  He  then  "unfolds 
some  other  reflections"  on*  two  regular  solids  the  dodecagon  and  icosahedron  "the  former  of  which  is  made 
up  entirely  of  pentagons,  the  latter  of  triangles  arranged  in  pentagonal  form.  The  structure  of  these  solids 
in  a  form  so  strikingly  pentagonal  could  not  come  to  pass  apart  from  that  proportion  which  geometers  to-day 
pronounce  divine."  In  discussing  this  divine  proportion  he  arrives  at  the  series  of  numbers  1,  1,  2,  3,  5,  8, 
13,  21  and  concludes:  "For  we  will  always  have  as  5  is  to  8  so  is  8  to  13,  practically,  and  as  8  is  to  13,  so  is 
13  to  21  almost.  I  think  that  the  seminal  faculty  is  developed  in  a  way  analogous  to  this  proportion  which 
perpetuates  itself,  and  so  in  the  flower  is  displayed  a  pentagonal  standard,  so  to  speak.  I  let  pass  all  other 
considerations  which  might  be  adduced  by  the  most  delightful  study  to  establish  this  truth." 

6  There  is  a  typographical  error  (13  for  21)  in  Girard's  discussion  in  this  connection. 

6  V.  Schlegel,  "Verallgemeinerung  eines  geometrischen  Paradoxons,"  Zeitschrijt  fiir  Mathematik  und 
Physik,  24.  Jahrgang,  1879,  pp.  123-128. 

7  This  paradox  was  given  at  least  as  early  as  1868  in  Zeitschrijt  fiir  Mathematik  und  Physik,  Vol.  13,  p. 
162.  Cf.  W.  W.  R.  Ball,  Mathematical  Recreations  and  Essays,  5th  edition,  London,  Macmillan,  191 1,  p.  53; 
and  E.  B.  Escott,  "Geometric  Puzzles,"  Open  Court  Magazine,  vol.  21,  1907,  pp.  502-5. 

8  E.  Catalan,  Melanges  Mathematiques,  tome  2,  [Liege,  1887],  p.  319. 


156 


DYNAMIC  SYMMETRY 


many  other  properties,  by  Lucas,1  in  a  paper  showing  the  relation  between  the  Fibonacci  series 
and  Pascal's  arithmetical  triangle.  Daniel  Bernoulli  showed2  that 


1  +  V5 


I-V5 


2      I         \  1 
from  this  a  result  given  by  Catalan  read 


V5; 

ly  follows:3 


n  ,     n{n  —  1)  (n  —  2)  nn{n 

-  +  5  "  b  52  - 

1  .1.2.3 


1)  (n  -  2)  («  -  3)  (n  -  4) 
1.2.3.4.5 


+ 


A  very  similar  series  occurs  in  a  letter  written  by  Euler  in  1726. 

Lucas  showed  the  importance  of  the  Fibonacci  series  in  discussions  of  {a)  the  decomposition 
of  large  numbers  into  factors  and  (b)  the  law  of  distribution  of  prime  numbers.4  Binet  was  led  to 
the  series  in  his  memoir  on  linear  difference  equations  (/.  c),  and  Leger6  and  Finck6  (and  later 
Lame')  indicated  its  application  in  determining  an  upper  limit  to  the  number  of  operations 
made  in  seeking  the  greatest  common  divisor  of  two  integers.  Landau  evaluated  the  series 
2(1/ uin  and  2(1/ «2n+i),  and  found  that  the  first  was  related  to  Lambert's  series  and  the  second 
to  the  theta  series.8 

The  solution  of  the  problem  of  determining  the  convex  polyhedra,  the  number  of  whose 
vertices,  faces,  and  edges  are  in  geometrical  progression,  leads  to  the  Fibonacci  series.9 

For  further  references  and  mathematical  discussions  one  may  consult  (1)  L'Intermediaire 
des  mathcmaticiens,  1899,  p.  242;  1900,  pp.  172-7,  251;  1901,  92;  1902,  p.  43;  1913,  pp.  50,  51, 


1  E.  Lucas,  "Note  sur  la  triangle  arithmetique  de  Pascal  et  sur  la  serie  de  Lame,"  Nouvelle  correspondance 
mathematique,  tome  2,  1876,  p.  74. 

2  D.  Bernoulli,  "Observationes  de  seriebus  quae  formantur  ex  additione  vel  subtractione  quacunque 
terminorum  se  mutus  consequentium,"  Commentarii  academiae  scientiarum  imperialis  Petropolitanae,  vol. 
3,  1732,  p.  90.  This  memoir  was  read  in  September,  1728,  but  it  appears  that  Bernoulli  had  the  formula  in 
his  possession  as  early  as  1724  (Cf.  Fuss,  Correspondance  mathematique  et  physique,  St.  Petersburg,  1843, 
vol.  2,  pp.  189,  193-4,  200-202,  209,  239,  251,  271,  277;  see  also  p.  710).  The  formula  was  given  also  by 
Euler  in  1726  (in  an  unpublished  letter  to  Daniel  Bernoulli).  For  most  of  these  facts  I  am  indebted  to  Mr. 
G.  Enestrom.  The  formula  seems  to  have  been  discovered  independently  by  J.  P.  M.  Binet,  "Memoire  sur 
l'integration  des  equations  lineaires  aux  differences  finies  d'un  ordre  quelconque,  a  coefficients  variables," 
Comptes  rendus  de  V academic  des  sciences  de  Paris,  tome  17,  1843,  P»  5^3- 

3  Manuel  des  Candidats  a  I'Ecole  Polytechnique,  tome  1,  Paris,  1857,  p.- 86. 

4  E.  Lucas,  {a)  "Recherches  sur  plusieurs  ouvrages  de  Leonard  de  Pise  et  sur  divers  es  questions  d'arith- 
metique  superieure.  Chapter  I.  Sur  les  series  recurrentes,"  Bullettino  di  bibliografia  e  di  storia  delle  scienze 
matematiche  e  fisiche,  tome  10,  pp.  129-170,  Marzo,  1877;  (b)  Theorie  des  fonctions  numeriques  simplement 
periodiques,"  American  Journal  oj  Mathematics,  vol.  r,  1878,  pp.  184-229,  289-321  [on  p.  299  are  given  the 
first  61  terms  of  the  Fibonacci  series  and  the  factors  of  every  term];  (c)  "Sur  la  theorie  des  nombres  premiers" 
[dated  mai  1876],  Atti  della  r.  accademia  delle  scienze  di  Torino,  vol.  II,  1875-76,  pp.  928-937;  (d)  "Note 
sur  l'application  des  series  recurrentes  a  la  recherche  de  la  loi  de  distribution  des  nombres  premiers,"  Comptes 
rendus  de  I'academie  des  sciences,  vol.  82,  1876,  pp.  165-167.  See  also  A.  Aubry,  "Sur  divers  procedes  de 
factorisation,"  L Enseignement  mathematique,  1913,  especially  §§  11,  16  and  17,  pp.  219-223. 

6  "Note  sur  le  partage  d'une  droite  en  moyenne  et  extreme,  et  sur  un  probleme  d'arithmetique,"  Corre- 
spondance mathematique  et  physique,  vol.  9,  1837,  pp.  483-484. 

6  Traite  Elementaire  d' Arithmetique,  Paris,  1841;  also  Nouvelles  annales  de  mathematique s,  vol.  1,  1842,  p. 
354. 

7  G.  Lame,  "Note  sur  la  limite  du  nombre  des  divisions  dans  la  recherche  du  plus  grand  commun  diviseur 
entre  deux  nombres  entiers."  Comptes  rendus  de  Facadimie  des  sciences,  tome  19,  1844,  pp.  867-870.  See 
also  J.  P.  M.  Binet,  idem,  pp.  939-941. 

Because  of  results  obtained  in  the  above-mentioned  memoir  the  Fibonacci  series  is  frequently  called  the 
Lame  series.  Thompson's  statement  {On  Growth  and  Form,  p.  643)  that  the  series  2/3,  3/5,  5/8,  8/13,  13/21, 
. . .  "is  called  Lami's  series  by  some,  after  Father  Bernard  Lami,  a  contemporary  of  Newton's,  and  one  of 
the  co-discoverers  of  the  parallelogram  of  forces,"  is  incorrect. 

8  E..  Landau,  "Sur  la  serie  des  inverses  des  nombres  de  Fibonacci,"  Bulletin  de  la  Societe  Mathematique  de 
France,  tome  27,  1899,  pp.  298-300. 

9  Archiv  der  Mathematik  und  Physik  Band  28,  1919,  pp.  77-79- 


DYNAMIC  SYMMETRY 


157 


147;  191 5,  pp.  39-40  (see  also  question  4171,  1915,  p.  277);  (2)  "Sur  une  generalisation  des 
progressions  geometriques,"  L'Education  mathematique,  1914,  pp.  149-151,  157-158;  (3)  V. 
Schlegel,  "Series  de  Lame  superieurs,"  El progreso  matematico,  1894,  ano  4,  pp.  171-174;  (4) 
T.  H.  Eagles,  Constructive  Geometry  of  Plane  Curves,  London,  1885,  pp.  293-299,  303-304;  and 
(5)  L.  E.  Dickson,  History  of  the  theory  of  Numbers,  vol.  1,  Washington,  1919,  Chapter  XVII: 
"Recurring  series;  Lucas'  un,  vn." 

As  to  growths  it  is  particularly  in  connection  with  older  chapters  on  leaf  arrangement  or 
phyllotaxis  that  the  Fibonacci  series  comes  up.  Among  the  earliest  and  most  important  of  these 
are  the  memoirs  of  Braun  (based  on  researches  of  Schimper  and  himself),1  and  L.  et  A.  Bravais.2 
Of  later  papers  there  are  those  by  Ellis,3  Dickson,4  Wright,5  Airy,6  Gunther,7  and  Ludwig.8 
Much  that  was  fanciful  and  mysterious  was  swept  away  by  the  publication  of  P.  G.  Tait's  note 
"On  Phyllotaxis."9  Of  recent  books  on  the  subject  the  most  notable  are  those  by  Church,10 
Cook,11  and  Thompson.12  The  first  two  are  beautifully  illustrated..  The  third  is  a  scholarly 
work,  written  in  an  attractive  style;  it  reproduces  Tait's  discussion  in  an  appreciative  manner. 

NOTE  VI. 

?  Tl  ^HIS  idea  of  commensurability  or  measurability  in  square  is  geometrically  explained 
in  the  tenth  book  of  Euclid's  "Elements."  The  artistic  use  of  this  fact  became  lost.  This 

J  t  loss  was  a  calamity.  We  must  either  blame  the  Romans  for  this  catastrophe  or  ascribe 
it  to  a  general  deterioration  of  intelligence.  If  this  knowledge  had  not  become  lost  artists  today 
would,  undoubtedly,  have  been  creating  masterpieces  of  statuary,  painting  and  architecture 
equalling  or  surpassing  the  masterpieces  of  the  Greek  classic  age. 

Since  the  material  for  this  book  was  obtained  the  writer  has  continued  the  work  of  analyses 
of  other  phases  of  Greek  design  such  as  that  furnished  by  the  temples,  bronzes,  stele  heads  and 
general  decoration.  To  this  has  been  added  a  close  inspection  of  the  architecture  of  man,  both 
in  the  skeleton  and  in  the  living  example;  and  the  human  figure  has  been  compared  with  Greek 
statuary.  The  results  of  this  more  recent  work  show  quite  clearly  that  the  symmetry  of  man,  as 
well  as  the  symmetry  of  Greek  statuary,  is  dynamic.  The  symmetry  of  the  human  figure  in 
art  since  the  first  century  B.  C.  is  undoubtedly  static.  From  the  fact  that  we  do  not  find  this  type 


1  A.  Braun,  "Vergleichende  Untersuchung  iiber  die  Ordnung  der  Schuppen  an  den  Tannenzapfen  als 
Einleitung  zur  Untersuchung  der  Blatterstellung  uberhaupt,"  Nova  acta  acad.  Caes  Leopoldina,  vol.  15, 
1830,  pp.  1 99-40 1. 

2  L.  et  A.  Bravais,  (1)  "Sur  la  disposition  des  feuilles  curviseriees,"  Ann.  des  sc.  nat.,  2e  serie,  vol.  7,  1837, 
pp.  42-110;  (2)  Memoire  sur  la  Disposition  geometrique  des  Feuilles  et  des  Inflorescenses,  Paris,  1838. 

3  R.  L.  Ellis,  Mathematical  and  Other  Writings,  Cambridge,  1863;  "On  the  theory  of  vegetable  spirals," 

PP-  358-372- 

4  A.  Dickson,  "On  some  abnormal  cases  of  pinus  pinaster,"  Transactions  of  the  Royal  Society  of  Edinburgh, 
vol.  26,  1 87 1,  pp.  505-520. 

6  C.  Wright,  "The  uses  and  origin  of  the  arrangements  of  leaves  in  plants"  (read  1871),  Memoirs  of  the 
American  Academy,  vol.  9,  part  2,  Cambridge,  Mass.,  p.  384^ 

6  H.  Airy,  "On  leaf  arrangement,"  Proceedings  of  the  Royal  Society  of  London,  vol.  21,  1873,  pp.  176-179. 

7  S.  Gunther,  "Das  mathematische  Grundgesetz  im  Bau  des  Pflanzenkorpers,"  Kosmos,  II.  Jahrgang, 
Band  4,  1879,  pp.  270-284. 

8  F.  Ludwig,  "Einige  wichtige  Abschnitte  aus  der  mathematischen  Botanik,"  Zeitschriftfur  mathematischen 
und  naturwiss.  Unterricht,  Band  14,  1883,  p.  i6if. 

9  P.  G.  Tait,  Proc.  Royal  Society  Edinburgh,  vol.  7,  1872,  pp.  391-4. 

10  A.  H.  Church,  On  the  Relation  of  Phyllotaxis  to  Mechanical  Laws,  London,  Williams  and  Norgate,  1904. 
On  page  5  Church  writes:  "The  properties  of  the  Schimper-Braun  series  I,  2,  3,  5,  8,  13,  .  .  .,  had  long  been 
recognized  by  mathematicians  (Gerhardt,  Lame).  .  .  ."  In  Botanisches  Centralblatt,  Band  68,  1896,  F.  Lud- 
wig writes  (on  p.  7)  that  the  numbers  of  this  series  "werden  vielfach  von  Botanikern  als  Braun'sche,  von 
Mathematikern  als  Gerhardt'sche  oder  Lame'sche  Reihe  bezeichnet."  I  have  not  been  able  to  verify  that 
any  mathematician  used  the  term  Gerhardt  series  in  this  connection,  or  that  anyone  by  the  name  of  Ger- 
hardt wrote  about  the  Fibonacci  series.  From  what  has  been  indicated  above  it  seems  certain  that  "Ger- 
hardt'sche" should  be  "Girard'sche." 

11  T.  A.  Cook,  The  Curves  of  Life,  London,  Constable,  1914. 

12  D'A.  W.  Thompson,  On  Growth  and  Form,  Cambridge:  at  the  University  Press,  1917. 


i58 


DYNAMIC  SYMMETRY 


of  symmetry  in  the  living  example  it  seems  fair  to  assume  that  static  man  could  not  function 
and,  therefore,  the  human  figure  in  art  of  the  past  two  thousand  years  is  not  true  to  nature. 

Since  the  publication  of  Darwin's  "Origin  of  Species,"  an  enormous  amount  of  human  meas- 
urement material  or  data  has  been  produced.  During  the  American  Civil  War  measurements 
were  obtained  of  over  a  million  recruits  and  drafted  men.  To  add  to  this  we  have  the  results  of 
the  activities  of  the  anthropologists  the  -world  over  during  the  past  generation.  All  this  data 
confirms  the  dynamic  hypothesis.  Since  the  first  century  B.  C.  many  treatises  have  been 
written  upon  the  proportions  of  the  human  figure  by  artists  and  others.  Bertram  Windle,  an 
English  lecturer  on  art,  has  prepared  a  table  of  some  eighty-eight  names.  To  this  we  may  add 
the  canons  of  proportion  used  in  the  continental  studios  during  the  past  hundred  years.  If 
human  figures  were  made  according  to  the  principles  enunciated  in  these  treatises  and 
canons,  the  result  would,  automatically,  be  static.  If  artists  made  human  figures  in  accordance 
with  the  measurements  obtained  by  anthropologists  and  by  the  different  governments,  of  men 
in  the  armies  and  navies,  the  result  would  also  be  static;  though  the  latter  would  be  truer  to 
nature  than  the  figures  made  according  to  the  artistic  canons,  because  men  of  science  have  found 
that  the  members  of  the  human  body  are  incommensurate;  to  meet  this  difficulty  they  use  a 
decimal  system.  This  is  nearer  nature  than  the  artists'  schemes  of  commensurate  length  units 
used  by  artists. 

One  reason  why  we  seem  to  have  failed  to  construct  the  human  figure  true  to  nature  appears 
to  be  due  to  Roman  misinterpretation  of  a  Greek  tradition  and  the  persistence  of  this  misin- 
terpretation through  the  ages  since.  The  tradition,  according  to  the  Roman  architectural  writer 
Vitruvius,  was  that  the  Greeks  based  the  symmetry  they  were  so  careful  to  apply  to  works  of 
art,  upon  the  commensurate  relationship  of  the  members  of  the  human  body  to  the  structure  as 
a  whole.  The  Romans  assumed  that  this  commensuration  or  measurableness  was  that  of  line. 
The  members  of  the  body  are,  indeed,  commensurable  or  measurable  with  the  structure  as  a 
whole,  but  in  area,  not  in  line. 

Greek  scientists  clearly  understood  that  lines  incommensurable  or  unmeasurable,  one  by  the 
other,  as  lengths,  were  not  necessarily  irrational;  they  might  be  commensurable  in  square.  Greek 
design  shows  that  Greek  artists  also  understood  this  fact. 

If  a  projection  is  made  of  the  living  model,  or  the  skeleton,  and  the  members,  such  as  the 
hands,  feet,  arms,  legs,  head,  trunk,  etc.,  be  compared  with  the  whole  in  terms  of  area  a  theme 
will  be  disclosed  and  this  theme  will  be  recognized  as  dynamic  exactly  as  are  the  area  themes  we 
obtain  from  a  Greek  temple  or,  indeed,  from  almost  any  example  of  good  Greek  design.  And 
such  themes  of  area  show  also  that  the  architecture  of  the  plant  and  that  of  man  are  essentially 
the  same. 

NOTE  VII. 


?  [~1  ^HE  reciprocal  idea,  especially  in  connection  with  design,  is  quite  unknown  to  modern 
artists.  It  was,  however,  well  understood  by  the  Greek  masters  as  their  design  creations 
abundantly  prove.  The  modern  mathematician  understands  the  value  of  the  reciprocal 
of  a  number  and  uses  it  to  shorten  certain  mathematical  operations.  For  example;  if  it  is  desired 
to  divide  one  number  by  another  the  same  result  is  obtained  if  that  number  be  multiplied  by 
the  reciprocal  of  the  other  number.  A  reciprocal  is  obtained  by  dividing  a  number  into  unity. 
.5  is  the  reciprocal  of  2.  and  any  number  multiplied  by  .5  produces  a  result  equivalent  to  dividing 
that  number  by  2.  In  this  example  simple  numbers  are  employed,  but  it  will  be  apparent 
that  a  problem  might  involve  a  very  complicated  and  unwieldy  number  and  in  that  case  the 
operation  would  be  much  simplified  if  multiplication  by  a  reciprocal  were  done  instead  of  division 
by  the  original  number.  This  valuable  property  of  the  reciprocal  forms  part  of  the  machinery 
of  dynamic  symmetry,  and  its  chief  use  is  that  of  determining  similar  figures  for  purposes  of 
design.  The  rectangular  shapes  derived  from  animal  or  plant  growth  may  all  be  expressed  by  a 
ratio.  This  fact  enables  us  to  perform  most  extraordinary  feats  of  design  analysis  by  simple 
arithmetic.  If  we  measure  a  Greek  design,  for  example,  and  find  that  it  is  contained  in  a  rectangle 
and  that  the  short  end  of  this  rectangle  divided  into  its  long  side  produces,  say,  the  ratio  2.236 


DYNAMIC  SYMMETRY 


159 


we  know  that  we  have  found  an  example  of  Greek  design  in  a  root-five  rectangle,  because  2.236 
is  the  square  root  of  five.  We  also  know  that  there  is  another  number  which  expresses  this  same 
fact  and  that  number  is  the  reciprocal  of  2.236.  To  obtain  this  reciprocal  we  divide  2.236 
into  unity:  the  answer  is  .4472.  Because  a  reciprocal  shape  is  a  similar  shape  to  the  whole  we 
know  that  .4472  also  represents  a  root-five  rectangle.  In  root  rectangles  the  reciprocal  is  always 
an  even  multiple  of  the  whole.  .4472  multiplied  by  5  equals  2.2360.  Consequently,  the  area  of 
a  root-five  rectangle  is  composed  of  five  reciprocal  areas.  As  a  labor-saver  the  property  of  the 
reciprocal  is  as  great  in  design  as  it  is  in  mathematics.  Also,  it  should  be  remembered  that 
reciprocal  ratios  are  always  less  than  unity.  Because  of  this  we  know  that  any  ratio  less  than 
unity  is  the  reciprocal  of  some  ratio  greater  than  unity.  Diagonals  to  reciprocals  always  cut  the 
diagonals  of  the  whole  at  right  angles. 

NOTE  VIII. 

ROOT-TWO  and  root-three  rectangles  never  appear  in  connection  with  root-five  and  the 
rectangle  of  the  whirling  squares.  For  this  reason  it  may  be  that  the  root-two  and  root- 
three  shapes  constitute  a  type  of  symmetry  intermediate  between  static  and  dynamic 
or  constitute  a  minor  phase  of  the  dynamic  type.  They  are  not  found  in  the  plant  or  the  human 
figure  or  in  Greek  statuary. 

NOTE  IX. 

?  T]  ^HE  summation  series  of  numbers  represents  an  extreme  and  mean  ratio  series  approx- 
imately, or  as  nearly  as  may  be  by  whole  numbers.  For  an  exact  representation  we  must 

„JL  use  a  substitute  series.  A  suggestion  for  such  a  substitute  series  is  furnished  by  the 
human  figure  and  Greek  design.  Such  a  series  would  be:  118  .  191  .  309  .  500  .  809  .  1309  .  21 18  . 
3427  .  5545  .  6854    8972  .  14517.,  etc.  _ 

Any  member  of  the  series  divided  into  any  succeeding  member  produces  the  ratio  1.618. 
Members  divided  into  alternate  members,  as  5  into  1309  produce  the  ratio  2.618. 

2.618  is  the  square  of  1.618,  that  is  1.618  multiplied  by  itself.  Also  1.618  plus  1  equals  1.618 
squared.  Every  member  divided  into  every  fourth  member  produces  the  ratio  4.236.  This  ratio 
equals  1.618  raised  to  the  third  power.  Also,  2.618  plus  1.618  equals  4.236.  Also  1.618  multiplied 
by  two  and  one  added  equals  4.236  and  so  on. 

NOTE  X. 

P  Tl  ^HE  root  rectangles  are  constructed  by  a  simple  geometrical  process.  The  instrument 
for  the  purpose  need  not  be  more  complicated  than  that  of  a  string  the  ends  of  which 

_  _L  are  held  in  the  two  hands.  The  constructions  depend  upon  the  Greek  method  of  determin- 
ing multiple  squares.  The  ancient  surveyor  being  called  a  "rope  stretcher,"  the  craftsman,  using 
the  same  method,  might  be  termed  a  "string  stretcher." 

"In  the  determination  of  a  square,  which  shall  be  any  multiple  of  the  square  on  the  linear  unit, 
a  problem  which  can  be  easily  solved  by  successive  applications  of  the  'theorem  of  Pythagoras' — ■ 
the  first  right-angled  triangle,  in  the  construction,  being  isosceles,  whose  equal  sides  are  the 
linear  unit;  the  second  having  for  sides  about  the  right  angle  the  hypotenuse  of  the  first  (root  1) 
and  the  linear  unit;  the  third  having  for  sides  about  the  right  angle  (root  3)  and  1,  and  for  hypot- 
enuse 2,  and  so  on."  Allman,  Greek  Geometry,  p.  24. 

"Theaetetus  relates  how  his  master  Theodorus,  who  was  subsequently  the  mathematical 
teacher  of  Plato,  had  been  writing  out  for  him  and  the  younger  Socrates  something  about 
squares;  about  the  squares  whose  areas  are  three  feet  and  five  feet  (these  squares  would  be 
those  on  the  sides  of  a  root-three  and  a  root-five  rectangle),  showing  that  in  length  they  are 
not  commensurable  with  the  square  whose  area  is  one  foot  (that  the  sides  of  the  square  whose 
areas  are  three  superficial  feet  and  five  superficial  feet  are  incommensurable  with  the  side  of  the 
square  whose  area  is  the  unit  of  surface,  i.  e.,  are  incommensurable  with  the  unit  of  length)  and 
that  Theodorus  had  taken  up  separately  each  square  as  far  as  that  whose  area  is  seventeen 


i6o 


DYNAMIC  SYMMETRY 


square  feet,  and,  somehow,  stopped  there.  Theaetetus  continues: — 'Then  this  sort  of  thing 
occurred  to  us,  since  the  squares  appear  to  be  infinite  in  number,  to  try  and  comprise  them  in 
one  term,  by  which  to  designate  all  these  squares.' 
"Socrates.  'Did  you  discover  anything  of  the  kind?' 

"Theaetetus.  'In  my  opinion  we  did.  Attend,  and  see  whether  you  agree.' 
"Socrates.  'Go  on.'  r 

"Theaetetus.  'We  divided  all  number  into  two  classes;  comparing  that  number  which  can  be 
produced  by  the  multiplication  of  equal  numbers  to  a  square  in  form,  we  called  it  quadrilateral 
and  equilateral.' 

"Socrates.  'Very  good.' 

"Theaetetus.  'The  numbers  which  lie  between  these,  such  as  three  and  five,  and  every  number 
which  cannot  be  produced  by  the  multiplication  of  equal  numbers,  but  becomes  either  a  larger 
number  taken  a  lesser  number  of  times,  or  a  lesser  taken  a  greater  number  of  times  (for  a  greater 
factor  and  a  less  always  compose  its  sides);  this  we  likened  to  an  oblong  figure,  and  called  it  an 
oblong  number.' 

"Socrates.  'Capital!  What  next?' 

"Theaetetus.  'The  lines  which  form  as  their  squares  an  equilateral  plane  (square)  number,  we 
defined  as  length,  i.  e.,  containing  a  certain  number  of  linear  units,  and  the  lines  which  form  as 
their  squares  an  oblong  number,  we  defined  as  dunameis,  inasmuch  as  they  have  no  common 
measure  with  the  former  in  length,  but  in  the  surfaces  of  the  squares,  which  are  equivalent  to 
these  oblong  numbers.  And  in  like  manner  with  solid  numbers.' 

"Socrates.  'The  best  thing  you  could  do,  my  boys;  no  one  could  do  better.'  "  Allman,  201-210. 
(These  boys  were  working  out  root-rectangles,  which  seem  to  have  been  familiar  to  the  elder 
Socrates,  who,  before  he  became  a  philosopher,  was  a  stone-cutter.) 

NOTE  XI. 

SEE  the  "Thirteen  Books  of  Euclid's  Elements"  by  Thomas  L.  Heath  and  his  reference 
to  Proclus. 

NOTE  XII. 

7   T]   ^HE  terms  "ellipse,"  "parabola"  and  "hyperbola"  were  first  used  in  connection  with 
this  process  of  the  "Application  of  Areas."  They  were  afterwards  applied  to  conic 
J  (     sections.  See  Heath. 


NOTE  XIII. 


""^HE  Parthenon  at  Athens  has  been  analyzed  by  dynamic  symmetry  and  the  proportions 
of  the  building  determined  to  the  minutest  detail.  The  theme  throughout  is  that  of 
J_L    square  and  root  five.  This  building,  and  other  Greek  temples,  are  examined  exhaustively 
in  monographs  now  in  preparation. 

NOTE  XIV. 


~^HE  connection  between  the  geometry  of  art  and  the  geometry  of  science  in  Greece  is 
shown  by  the  history  of  the  "Duplication  of  the  cube  problem."  In  Greece,  as  in  India, 
the  geometry  of  art  was  used  in  architecture  very  early.  In  the  former  it  is  the 
Delian  or  duplication  problem,  in  the  latter  "the  rules  of  the  chord,"  both  ideas  being  involved 
in  altar  ritual.  The  Greeks  reduced  the  duplication  problem  to  one  of  finding  two  mean  propor- 
tionals between  two  lines.  The  artist  uses  the  inverse  of  this  idea  in  dynamic  symmetry;  he  is 
constantly  dealing  with  two  mean  proportionals  between  two  lines.  Allman's  suggestion  that 
the  problem  arose  in  the  needs  of  architecture  is  undoubtedly  correct.  The  duplication  of  the 
cube  problem  arose  naturally  from  the  duplication  of  the  square. 

"The  Pythagoreans,  as  we  have  seen,  had  shown  how  to  determine  a  square  whose  area  was 


DYNAMIC  SYMMETRY 


161 


any  multiple  of  a  given  square.  The  question  now  was  to  extend  this  to  the  cube,  and,  in  par- 
ticular, to  solve  the  problem  of  the  duplication  of  the  cube."  Allman,  "History  of  Greek  Geom- 
etry from  Thales  to  Euclid,"  pp.  83-84. 

THE  DUPLICATION  OF  THE  CUBE 

ROCLUS  (after  Eudemus)  and  Eratosthenes  tell  us  that  Hippocrates  reduced  this 
question  ('the  duplication  of  the  cube')  to  one  of  plane  geometry,  namely,  the  finding 
of  two  mean  proportionals  between  two  given  straight  lines,  the  greater  of  which  is 
double  the  less.  Hippocrates,  therefore,  must  have  known  that  if  four  straight  lines  are  in  con- 
tinued proportion,  the  first  has  the  same  ratio  to  the  fourth  that  the  cube  described  on  the  first, 
as  side,  has  to  the  cube  described  in  like  manner  on  the  second.  He  must  then  have  pursued  the 
following  train  of  reasoning: — Suppose  the  problem  solved,  and  that  a  cube  is  found  which  is 
double  the  given  cube;  find  a  third  proportional  to  the  sides  of  the  two  cubes,  and  then  find  a 
fourth  proportional  to  these  three  lines;  the  fourth  proportional  must  be  double  the  side  of  the 
given  cube;  if,  then,  two  mean  proportionals  can  be  found  between  the  side  of  the  given  cube  and 
a  line  whose  length  is  double  of  that  side,  the  problem  will  be  solved.  As  the  Pythagoreans  had 
already  solved  the  problem  of  finding  a  mean  proportional  between  two  given  lines, — or,  which 
comes  to  the  same,  to  construct  a  square  which  shall  be  equal  to  a  given  rectangle — it  was  not 
unreasonable  for  Hippocrates  to  suppose  that  he  had  put  the  problem  of  the  duplication  of  the 
cube  in  a  fair  way  of  solution.  Thus  arose  the  famous  problem  of  finding  two  mean  proportionals 
between  two  given  lines — a  problem  which  occupied  the  attention  of  geometers  for  many  cen- 
turies." Allman,  p.  84. 

We  must  not  forget  that  conic  sections  were  discovered  while  a  great  Greek  geometer  was 
trying  to  solve  this  problem  of  two  mean  proportionals. 

Plutarch,  Life  of  Marcellus:  "  'The  first  who  gave  an  impulse  to  the  study  of  mechanics,  a 
branch  of  knowledge  so  prepossessing  and  celebrated,  were  Eudoxus  and  Archytas,  who  em- 
bellish geometry  by  means  of  an  element  of  easy  elegance,  and  underprop,  by  actual  experiments 
and  the  use  of  instruments,  some  problems  which  are  not  well  supplied  with  proof  by  means  of 
abstract  reasonings  and  diagrams;  that  problem,  for  example,  of  two  mean  proportional  lines, 
which  is  also  an  indispensable  element  in  many  drawings'  "  Allman  p.  159. 

"Eratosthenes,  in  his  letter  to  Ptolemy  III,  relates  that  one  of  the  old  tragic  poets  introduced 
Minos  on  the  stage  erecting  a  tomb  for  his  son  Glaucus;  and  then,  deeming  the  structure  too 
mean  for  a  royal  tomb,  he  said;  'double  it  but  preserve  the  cubical  form.'  Eratosthenes  then 
relates  the  part  taken  by  Hippocrates  of  Chios  towards  the  solution  of  this  problem  and 
continues  'Later  (in  the  time  of  Plato),  so  the  story  goes,  the  Delians,  who  were  suffering  from  a 
pestilence,  being  ordered  by  the  oracle  to  double  one  of  their  altars,  were  thus  placed  in  the  same 
difficulty.  They  sent,  therefore,  to  the  geometers  of  the  Academy,  entreating  them  to  solve  the 
question.'  This  problem  of  the  duplication  of  the  cube,  henceforth  known  as  the  Delian  Problem, 
may  have  been  originally  suggested  by  the  practical  needs  of  architecture,  as  indicated  in  the 
legend,  and  have  arisen  in  Theocratic  times;  it  may  subsequently  have  engaged  the  attention 
of  the  Pythagoreans  as  an  object  of  theoretic  interest  and  scientific  enquiry,  as  suggested  above." 
Allman,  p.  85. 


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